UC-NRLF 


7DM 


UNIVERSITY  OF  CALIFORNIA. 


FROM   THE    LIBRARY  OF 

DR.  JOSEPH   LECONTE. 

GIFT  OF  MRS.   LECONTE. 
No. 


, 


LECTURE-NOTES 


PHYSICS. 


BY  ALFKED  M.  MAYEK,  PHI>., 

PROFESSOK  OF  PHYSICS  IN  THE  LEHIGH  UNIVERSITY,  BETHLEHEM,  PA. 


PART  I.— Containing 

\  I.      DEFINITIONS  AND  INTRODUCTION  TO  THE  INDUCTIVE  METHOD. 

§  II.     INSTRUMENTS  USED  IN  PRECISE  MEASUREMENTS. 

$  III.  METHODS  OF  PRECISION. 

§  IV.  MANNERS  OF  EXPRESSING  A  LA.W — LAW  EVOLVED  FROM  THE  NUMERICAL 

RESULTS  OF  OBSERVATIONS  AND  EXPERIMENTS. 

\  V.     THE  GENERAL  PROPERTIES  OF  MATTER — THE  CONSTITUTION  OF  MATTER 

ACCORDING  TO  THE  MOLECULAR  HYPOTHESIS. 

2  VI.  CAPILLARY  ATTRACTION. 


PHILADELPHIA: 
FROM  THE  JOURNAL  OF  THE  FRANKLIN  INSTITUTE. 

1868. 


"Who  hath  measured  the  waters  in  the  hollow  of  His  hand,  and  meted  out  heaven 
with  the  span,  and  comprehended  the  dust  of  the  earth  in  a  measure,  and  weighed 
the  mountains  in  scales,  and  the  hills  in  a  balance?"  (!SAIAH  xl.  12.) 

"  Thou  hast  ordered  all  things  in  measure  and  number  and  weight."  (BooK  OF 
THE  WISDOM  OF  SOLOMON,  chap.  xi.  20.) 

"  It  ought  to  be  eternally  resolved  and  settled  that  the  understanding  cannot  decide 
otherwise  than  by  induction,  and  by  a  legitimate  form  of  it." 

"  Francis  of  Verulam  thought  thus,  and  such  is  the  method  which  he  determined 
within  himself,  and  which  he  thought  it  concerned  the  living  and  posterity  to 
know." 

"  Let  no  one  enter  here  who  is  ignorant  of  geometry."     (PLATO.) 


LECTURE-NOTES  ON  PHYSICS. 


§  I.  Definitions  and  Introduction  to  the  Inductive  Method. 

Physical  science  is  the  knowledge  of  the  laws  (c)  of  the  phenomena 
(b)  of  matter  (a). 

(a.)  Matter  is  that  which  affects  the  senses.  It  always  presents 
the  three  dimensions,  or,  in  other  words,  occupies  space. 

It  exists  throughout  all  known  space  as  a  highly  elastic  and  rare 
medium  called  ether  (proof  of  this  given  in  optics),  and  in  this  medium 
circulate  dense  bodies  of  spheroid  forms,  separated  from  each  other  by 
distances  which  are  immense  when  compared  with  the  size  of  these 
bodies;  such  are  the  planets  and  asteroids  of  celestial  space,  and  the 
earth  on  which  we  live. 

Shooting-stars  or  Aerolites  are  celestial  bodies  of  smaller  size, 
which  at  certain  periods  (November  13th  and  August  10th)  cross 
the  orbit  of  the  earth. 

Comets  are  highly  rarified  nebulous  bodies  of  various  aud  chang- 
ing forms,  circulating  in  orbits  which  are  so  eccentric,  that  they  are 
not  visible  to  us  in  their  remote  situations;  while  at  other  times  they 
are  much  nearer  to  the  sun  than  any  of  the  planets.  The  comet  of 
1680,  when  nearest  to  the  sun,  was  only  one-sixth  of  the  sun's  dia- 
meter from  his  surface. 

The  motion  of  comets  retarded  by  the  resistance  of  the  ether? 

The  above  is  a  statement  of  all  known  matter,  and  physical  science 
is  that  branch  of  knowledge  which  considers  all  the  various  pheno- 
mena which  this  matter  presents. 

From  the  investigations  of  chemists  on  terrestrial  matter,  and  from 
the  spectroscopic  examination  of  celestial  bodies,  all  matter  can  be 
resolved  into  (at  present)  sixty -three  elements. 

According  to  the  atomic  theory  see  (§V.)  all  matter  consists  of 
exceedingly  minute,  absolutely  jiard  and  unchangeable  atoms,  sepa- 
rated from  each  other  by  distances  which  are  very  great  when  com- 
pared with  the  dimensions  of  these  atoms, 

As  in  interplanetary  space  exists  the  rare  ether,  so  interatomic 

(3) 

145274 


4  Lecture- Notes  on  Physics. 

space  is  occupied  by  the  same  elastic  medium.  (Proofs  of  above  given 
under  Undulatory  Theory  of  Light  and  Heat.)  Matter  exists  in  the 
three  states  of  solids,  liquids,  and  gases. 

Each  organ  of  our  senses  is  so  constructed  that  it  can  only  take  cog- 
nizance of  those  effects  which  it  was  specially  designed  to  receive. 
Thus,  through  the  eye  we  perceive  light,  but  not  sound ;  and  the  ear 
does  not  take  cognizance  of  light,  nor  of  flavor,  nor  of  odor. 

All  the  senses  are  modifications  of  touch. 

In  touch,  taste,  and  smell  the  organs  of  these  senses  come  in  con- 
tact with  the  matter  that  we  touch,  taste,  and  smell.  In  the  cases  of 
sound,  of  light,  and  of  heat,  these  effects  are  propagated  through  an 
intervening  elastic  medium  whose  particles,  vibrating  in  unison  with 
the  particles  of  the  sounding,  luminous,  or  heated  body,  cause  the 
nerves  of  the  ear,  of  the  eye,  or  of  the  skin  to  pulsate;  and  thus  we 
have  special  sensations  which  we  severally  interpret  as  sound,  light, 
and  heat. 

Experiments  showing  the  transmission  of  sound — vibrations 
through  the  air,  through  a  liquid  and  through  a  long  rod  to  the 
ear.  The  human  body  can  be  the  intervening  vibrating  medium  to 
transmit  the  sound. 

Experiment.  Sound  not  propagated  through  a  vacuum.  Ether  is 
the  intervening  vibrating  medium  in  the  case  of  light  and  heat. 

A  shadow  is  not  matter,  because  it  does  not  affect  the  senses.  The 
light  reflected  from  the  surface,  contiguous  to  the  shadow,  affects  the 
eye;  but  from  where  the  shadow  is,  no  effect  emanates,  and  the  con- 
sciousness of  the  absence  of  an  effect  on  that  part  of  the  eye  where  the 
shadow  is  projected  gives  us  the  idea  of  darkness. 

According  to  Helmholtz  and  Du  Bois  Raymond  an  effect  is  propa- 
gated along  the  nerves  with  a  velocity  of  93  feet  in  one  second. 

(b.)  Phenomena.     A  collection  of  associated  facts. 

Example.  Compression  of  air.  Here  the  associated  facts  are  two. 
(1)  The  volumes  of  air  corresponding  to  (2)  different  pressures.  The 
fall  of  a  stone,  associated  fa-cts.  (1)  Spaces  fallen  through,  and  (2) 
the  times  occupied  in  falling  through  the  spaces. 

Every  physical  phenomenon  is  either  motion  or  the ^  result  of 
motion. 

Examples.  In  the  phenomena  of  astronomy,  mechanics,  acoustics, 
light,  heat,  electricity,  chemistry,  botany,  zoology. 

The  idea  of  motion  necessarily  embraces  two  others  viz:  that  of 
space  and  that  of  time;  for  a  motion  must  take  place  in  space,  and 


~ 

Lecture- Notes  on  Physics.  5 

must  happen  with  more  or  less  rapidity,  whence  our  idea  of  time.  (See 
Instruments  used  to  Measure  Time,  under  §  II.) 

(c.)  Law.  The  expression  of  the  general  relation  which  pervades 
a  class  of  facts,  or  the  rule  according  to  which  a  cause  acts. 

Examples.  The  law  of  the  compression  of  gases;  volumes  of  gases 
are  inversely  as  the  pressures  to  which  they  are  subjected.  Gravita- 
tion; the  gravitating  effect  is  directly  as  the  masses,  and  inversely 
as  the  square  of  the  distance  between  the  gravitating  bodies.  Sound, 
light,  heat,  electric  and  magnetic  attraction  diminish  in  intensity  in- 
versely as  the  squares  of  the  distances  from  their  centres  of  origin. 
Eeflection  and  refraction  of  light. 

Universality  of  these  general  relations  throughout  the  universe. 
Importance  of  these  general  expressions  in  assisting  the  memory.  A 
law  embraces  in  itself  all  the  myriads  of  facts  of  which  it  is  the  gene- 
ralization. 

The  use  of  a  law  is  shown  in  its  application  to  the  practical  pur- 
poses of  life. 

Examples.  Application  of  the  law  of  the  compression  of  gases  ;  of 
the  reflection  and  refraction  of  light ;  of  the  pressure  of  steam  at  dif- 
ferent temperatures ;  of  the  laws  of  dynamic  electricity. 

Without  the  knowledge  of  the  law  of  a  class  of  facts,  we  are  obliged 
to  make  experiments  to  solve  each  individual  problem,  while  if  we 
have  the  knowledge  of  the  law,  the  solution  of  the  problem  is  a  mere 
deduction  from  the  law  to  which  it  belongs  Examples  drawn  from, 
above  laws. 

The  money  value  of  a  law  to  the  engineer,  and  to  the  industrial 
and  commercial  world. 

The  discovery  of  laws  is  the  object  of  science. 

The  logical  arrangement  of  all  the  known  physical  laws  consti- 
tutes physical  science. 

"  Classification  of  the  Physical  Sciences. 

I.   INORGANIC.  II.   ORGANIC, 

a.  Celestial  Phenomena. 

1.  Astronomy.  1.  Botany. 

2.  Meteorology  (part  of )»  -2,  Zoology." 

b.  Terrestrial  Phenomena. 

1.  Physics. 

2.  Chemistry. 

3.  Geology,  including  Mineralogy* 

PROF.  J.  HENRY. 


6  Lecture- Notes  on  Physics. 

Physics  considers  the  laws  of  the  general  phenomena  of  matter,  or 
is  the  study  of  the  laws  of  those  phenomena  which  do  not  bring  about 
a  permanent  change  in  the  constitution  of  bodies. 

Examples.  Fall  of  a  stone,  compression  of  gases,  expansion  of 
bodies  bj  heat,  reflection  and  refraction  of  light,  &c. 

Chemistry  considers  the  peculiar  or  individual  phenomena  of  bodies, 
or  is  that  science  which  studies  those  phenomena  which  bring  about  a 
permanent  change  in  the  nature  of  bodies. 

Examples.  Burning  of  a  candle,  rusting  of  iron,  union  of  sulphur 
and  iron,  union  of  oxygen  and  hydrogen. 

The  fundamental  principle  on  which  we  repose  in  all  our  scientific 
reasoning  is,  that  the  laws  of  nature  are  constant,  or,  in  other  words, 
that  like  causes  will  always  produce  like  effects. 

"The  mode  of  reasoning,  in  physical  science,  which  is  the  most 
generally  to  be  adopted,  depends  on  this  axiom  which  has  always  been 
essentially  concerned  in  every  improvement  of  natural  philosophy,  but 
which  has  been  more  and  more  employed  ever  since  the  revival  of 
letters,  under  the  name  of  induction,  and  which  has  been  sufficiently 
discussed  by  modern  metaphysicians.  That  like  causes  produce  like 
effects,  or  that  in  similar  circumstances  similar  consequences  ensue,  is 
the  most  general  and  most  important  law  of  nature ;  it  is  the  foundation 
of  all  analogical  reasoning,  and  is  collected  from  constant  experience 
by  an  indispensable  and  unavoidable  propensity  of  the  human  mind." 
(Dr.  Thomas  Young's  Lectures  on  Natural  Philosophy,  Lecture  I., 
page  11.  London,  1845). 

"Most  of  the  phenomena  of  nature  are  presented  to  us  as  the  com- 
plex results  of  the  operation  of  a  number  of  laws.1' 

Examples^  In  astronomy,  in  souncl^  in  light,  in  heat,  and  in  elec- 
tricity. 

""  We  are  said  to  explain  or  give  the  cause  of  a  simple  fact  when  we 
refer  it  to  the  law  of  the  phenomena  to  which  it  belongs,  or  to  a  more 
general  fact ;  and  a  compound  one  when  we  analyze  it  and  refer  its 
•several  parts  to  their  respective  laws. 

u  The  indefinite  use  of  the  term  cause  has  led  to  much  confusion  and 
'error.  We  distinguish  two  kinds  of  causes,  intelligent  and  physical- 

"  By  an  intelligent  cause  is  meant  the  volition  of  an  intelligent  and 
efficient  being  producing  a  definite  result. 

"  By  a  physical  cause,  scientifically  speaking,  nothing  more  is  un- 
stood  than  the  law  to  which  a  phenomenon  can  be  referred. 

41  Thus  we  give  the  physical  cause  of  the  fall  of  a  stone  or  of  the  ele- 


Lecture- Notes  on  Physics.  7 

vation  of  the  tides,  when  we  refer  these  phenomena  to  the  law  of  gra- 
vitation ;  and  the  intelligent  cause  (sometimes  also  called  the  moral 
or  efficient  cause),  when  we  refer  this  law  to  the  volition  of  the  Deity." 

[It  is  generally  easy  to  know  to  what  physical  cause  we  should  refer 
a  class  of  physical  phenomena ;  but  what  is  often  difficult,  is  to  de- 
termine the  nature  and  manner  of  acting  of  that  cause,  and  to  show 
how  the  phenomena  and  their  laws  are  consequences  of  its  properties. 
This  is  the  nicest  point  in  physical  science.] 

"  In  the  investigation  of  the  order  of  nature,  two  general  methods 
have  been  proposed — I.  The  d  priori  method,  and  II.  The  inductive 
method. 

"  I.  The  d  priori  method  consists  in  reasoning  downwards  from  the 
original  cognitions,  which,  according  to  the  d  priori  philosophy,  exist 
in  the  mind  relative  to  the  nature  of  things,  to  the  laws  and  the  phe- 
nomena of  the  material  universe. " 

[Examples  given  of  ancient  and  medieval  d  priori  reasoning  in  their 
explanations  of  the  motions  of  the  heavenly  bodies,  and  of  the  law 
of  falling  bodies.] 

"  II.  The  inductive  method,  which  is  the  inverse  of  the  other,  is 
founded  on  the  principle  that  all  our  knowledge  of  nature  must  be 
derived  from  experience.  It  therefore  commences  with  the  study  of 
phenomena,  and  ascends  from  these  by  what  is  called  the  inductive 
process,  to  a  knowledge  of  the  laws  of  nature.  It  is  by  this  method 
that  the  great  system  of  modern  physical  science  has  been  established. 
It  was  used  in  a  limited  degree  by  the  ancients,  and  especially  by 
Aristotle,  but  its  importance  was  never  placed  in  a  conspicuous  light 
until  the  publication  of  the  Novum  Organum  of  Bacon  in  1620." 

"In  the  application  of  the  inductive  method  to  the  discovery  of 
the  laws  of  nature,  four  processes  are  usually  employed. 

"  1.  Observation,  which  consists  in  the  accumulation  of  facts,  by 
watching  the  operations  of  nature  as  they  spontaneously  present  them- 
selves  to  our  view. 

"  This  is  a  slow  process,  but  it  is  almost  the  only  one  which  can  be 
employed  in  some  branches  of  science, — for  example,  in  astronomy. 

"2.  Experiment,  which  is  another  method  of  observation,  in  which 
we  bring  about,  as  it  were,  a  new  process  of  nature  by  placing  matter 
in  some  new  condition." 

[In  making  an  experiment  we  should  not  only  evolve  the  phenomena 
whose  laws  we  wish  to  determine,  but  should  a] so  so  produce  these 
phenomena  that  we  can  measure  their  related  parts.] 


8  Lecture-Notes  on  Physics. 

"  This  is  a  much  more  expeditious  process  than  that  of  simple  ob- 
servation, and  has  been  aptly  styled  the  method  of  cross- questioning 
or  interrogating  nature. 

"The  term  experience  is  often  used  to  denote  either  observation, 
or  experiment,  or  both. 

["  A  most  important  remark,  due  to  Herschel,  regards  what  are 
called  residual  phenomena.  When,  in  an  experiment,  all  known 
causes  being  allowed  for,  there  remain  certain  unexplained  effects 
(excessively  slight  it  may  be),  these  must  be  carefully  investigated, 
and  every  conceivable  variation  of  arrangement  of  apparatus  etc. 
tried;  until,. if  possible,  we  manage  so  to  exaggerate  the  residual  phe- 
nomenon as  to  be  able  to  detect  its  cause.  It  is  here,  perhaps,  that  in 
the  present  state  of  science  we  may  most  reasonably  look  for  exten- 
sions of  our  knowledge  ;  at  all  events,  we  are  warranted  by  the  recent 
history  of  Natural  Philosophy  in  so  doing.  Thus,  for  example,  the 
slight  anomalies  observed  in  the  motion  of  Uranus  led  Adams  and 
Le  Yerrier  to  the  discovery  of  a  new  planet ;  and  the  fact  that  a  mag- 
netized needle  comes  to  rest  sooner  when  vibrating  above  a  copper 
plate  than  when  the  latter  is  removed,  led  Arago  to  what  was  once 
called  magnetism  of  rotation,  but  has  since  been  explained,  immensely 
extended,  and  applied  to  most  important  purposes.  In  fact,  this  acci- 
dental remark  about  the  oscillation  of  a  needle  led  to  facts  from 
which,  in  Faraday's  hands,  was  evolved  the  grand  discovery  of  the 
Induction  of  Electrical  Currents  by  magnets  or  by  other  currents." 
Natural  Philosophy,  by  Sir  William  Thomson  and  P.  G.  Tait.  Ox- 
ford, 1867,  p.  308.] 

"3.  The  Inductive  Process,  or  that  by  which  a  general  law  is  in- 
ferred from  particular  facts.  This  consists  generally  in  making  a 
number  of  suppositions  or  guesses  as  to  the  nature  of  the  law  to  be 
discovered,  and  adopting  the  one  which  agrees  with  the  facts.  The 
law  thus  adopted  is  usually  further  verified  by  making  deductions  from 
it  and  testing  these  by  experiment.  If  the  result  is  not  what  was 
anticipated,  the  expression  of  the  law  is  modified,  perhaps  many  times 
in  succession,  until  all  the  inferences  from  it  are  found  in  accordance 
with  the  facts  of  experience. " 

[Examples  of  induction  in  the  discovery,  by  Newton,  of  the  law  of 
gravitation;  of  the  composition  of  light;  of  Wells's  theory  of  dew;  of 
Ampere's  laws  of  electro-dynamics.] 

"4.  Deduction,  which  is  the  inverse  of  induction,  consists  in  rea- 
soning downwards  from  a  law,  which  has  been  established  by  indue- 


Lecture- Notes  on  Physics.  9 

tion,  to  a  system  of  new  facts.  In  this  process  the  strict  logic  of  mathe- 
matics is  employed,  the  laws  furnished  by  induction  standing  in  the 
place  of  axioms.  Thus  all  the  facts  relative  to  the  movements  of  the 
heavenly  bodies  have  been  derived,  by  mathematical  reasoning,  from 
the  laws  of  motion  and  universal  .gravitation."  (Prof.  J.  Henry.) 

[Examples  of  deduction.  The  discovery  of  the  planet  Neptune  by 
Le  Verrier  and  Adams ;  of  conical  refraction  by  Hamilton  and  Lloyd. 
Laplace's  work  Mechanique  Celeste^] 

"  Bacon's  greatest  merit  cannot  consist,  as  we  are  so  often  told 
that  it  did,  in  exploding  the  vicious  method  pursued  by  the  ancients 
of  flying  to  the  highest  generalizations  for  it,  and  deducing  the  middle 
principles  from  them ;  since  this  is  neither  a  vicious  nor  an  exploded 
method,  but  the  universally  accredited  method  of  modern  science,  and 
that  to  which  it  owes  its  greatest  triumphs.  The  error  of  ancient 
speculation  did  not  consist  in  making  the  largest  generalizations  first, 
but  in  making  them  without  the  aid  or  warrant  of  rigorous  inductive 
methods,  and  applying  them  deductively  without  the  needful  use  of 
that  important  part  of  the  deductive  method  termed  verification," 
(J.  Mills's  System  of  Logic,  vol.  ii.,  page  524-5.) 

The  agreement  of  the  deductions  which  we  make  from  a  law,  with 
the  facts  of  subsequent  observations  and  experiments,  is  the  only  test 
of  our  having  truly  arrived  at  a  law. 

Bacon  well  remarked,  that  the  test  of  true  science  was  its  utility, 
and  Franklin  was  well  aware  of  this  when  a  friend,  asking  him  the 
use  of  electricity,  he  replied,  characteristically,  "What  is  the  use  of  a 
baby?" 

"When  one  system  of  facts  is  similar  to  another,  and  when,  there- 
fore, we  infer  that  the  law  of  the  one  is  similar  to  the  law  of  the  other, 
we  are  said  to  reason  from  analogy. 

"  This  kind  of  reasoning  is  of  constant  use  in  the  process  of  induc- 
tion, and  is  founded  on  our  conviction  of  the  uniformity  of  the  laws  of 
nature. 

"In  the  process  of  the  discovery  of  a  law,  the  supposition  which  we 
make  as  to  its  nature  must  be  founded  on  a  physical  analogy  between 
the  facts  under  investigation  and  some  other  facts  of  which  the  law  is 
known.  One  successful  induction  is  thus  the  key  to  another. " 

[Examples.  Young's  discovery  of  the  interference  of  light ;  Frank- 
lin's of  the  identity  of  lightning  and  electricity.] 

"A  supposition  or  guess  thus  made  from  analogy  as  to  the  nature 
of  the  law  of  a  class  of  facts,  is  usually  called  an  hypothesis  and  some- 
times the  antecedent  probability. 


10  Lecture- Notes  on  Physics. 

"When  an  hypothesis  of  this  kind  has  been  extended  and  verified, 
or,  in  other  words,  when  it  has  become  an  exact  expression  of  the  law 
of  a  class  of  facts,  it  ig  then  called  a  theory. 

[Flourens,  in  his  Eloge  sur  Buff  on,  says :  "  An  hypothesis  is  the 
explanation  of  facts  by  possible  causes  ;  a  theory  is  the  explanation 
of  facts  by  real  causes. "] 

"  Physical  theories  are  of  two  kinds,  which  are  sometimes  called 
pure  and  hypothetical.  The  one  being  simply  the  expression  of  a 
law,  which  is  the  result  of  a  wide  induction,  resting  on  experiment  and 
observation.  Such  is  the  theory  of  universal  gravitation ;  the  theory 
of  sound. 

"  The  other  consists  of  an  hypothesis  combined  with  the  facts  of  ex- 
perience Of  this  kind  is  the  theory  of  electricity  which  attributes  a 
large  class  of  phenomena  to  the  operations  of  an  hypothetical  fluid 
endowed  with  properties,  so  imagined  as  to  render  the  theory  an  ex- 
pression of  the  law  of  the  facts.  "  (Prof.  J.  Henry.) 

A  theory  is  therefore  an  assemblage  of  the  facts  and  of  the  laws, 
with  the  consequences  which  may  therefrom  be  deduced,  which  be- 
long to  one  and  the  same  physical  cause.  Thus  we  have  the  theory 
of  gravitation ;  of  light ;  of  heat ;  of  sound,  &c. 

A  theory,  to  be  true  and  complete,  should  explain  to  their  minut- 
est details  all  the  phenomena  produced  by  the  cause,  and  that, 
naturally,  without  its  being  necessary  to  modify  the  given  facts  and 
their  laws.  Also  the  numerical  results  deduced  by  mathematical 
calculation  should  agree  with  those  furnished  by  direct  observation. 
If,  moreover,  the  logical  deductions  which  we  obtain  enable  us  to  pre- 
dict new  phenomena  which  experience  afterwards  confirms,  that 
theory  carries  with  it  the  most  convincing  proof  of  its  reality.  The 
theory  of  gravitation  and  the  undulatory  theory  of  light  afford  us 
beautiful  and  instructive  examples. 

In  order  to  establish  a  theory  and  follow  logically  the  consequences 
of  the  principle  from  which  we  have  started,  we  are  obliged  to  com- 
pare among  themselves  quantities  which  are  often  bound  together  by 
very  complicated  relations.  The  unaided  attention  is  not  equal  to 
such  un  effort,  and  hence  we  continually  call  in  the  aid  of  mathema- 
tical analysis  to  assist  us  in  our  investigations  in  physics. 

"  Strictly  speaking,  no  theory  in  the  present  state  of  science  can  be 
considered  as  an  actual  expression  of  the  truth. 

44  It  may,  indeed,  be  the  exact  expression  of  the  laws  of  a  limited 
class  of  facts,  but  in  the  advance  of  science  it  is  liable  to  be  merged 
in  a  higher  generalization  or  the  expression  of  a  wider  law. 


Lecture- Notes  on  Physics.  11 

"  It  should  be  recollected  that  the  laws  of  nature  are  contingent 
truths,  or  such  as  might  be  different  from  what  they  are,  for  anything 
we  know — that  they  can  only  be  established  by  induction  from  the 
facts  of  experience — that  they  admit  of  no  other  proof  than  the  d 
posteriori  one  of  the  exact  agreement  of  all  the  deductions  from  them 
with  the  actual  phenomena  of  nature,  and  that  no  other  cause  can 
be  assigned  for  their  existence  than  the  will  of  the  Creator. 

"  The  ultimate  tendency  of  the  study  of  the  physical  sciences  is  the 
improvement  of  the  intellectual,  moral,  and  physical  condition  of  our 
species.  It  habituates  the  mind  to  the  contemplation  and  discovery 
of  truth.  It  unfolds  the  magnificence,  the  order,  and  the  beauty  of 
the  material  universe,  and  affords  striking  proofs  of  the  beneficence, 
the  wisdom,  and  power  of  the  Creator.  It  enables  man  to  control  the 
operations  of  nature,  and  to  subject  them  to  his  use."  (Prof.  J. 
Henry.)* 

§  II.  Instruments  used  in  Precise  Measurements. 

A  law  of  physics,  as  we  have  seen,  consists  in  the  general  rela- 
tion which  exists  between  the  numbers  which  are  the  final  results 
of  our  observations  and  experiments;  and  therefore,  the  first  sub- 
ject to  be  discussed  in  the  exposition  of  that  process  by  which  we 
arrive  at  a  law,  is  the  description  of  those  instruments  which  have 
given  such  minute  precision  in  the  numerical  determinations  of 
modern  physicists. 

The  instruments  of  precision  used  in  making  the  measures  required 
by  the  physicist  maybe  arranged  under  the  five  following  divisions: — 

1.  Instruments  used  to  measure          .         .     Lengths. 

2.  "  "  .  Angles. 

3.  "  "  .  Volumes. 

4.  "  "  .  Weight. 

5.  "  «  .  Time. 

1.  Instruments  used  to  measure  Lengths — Dividing  ^Engines, 

Standards  of  Length.  Before  commencing  a  measurement  of  length, 
the  error  of  the  unit  we  use  should  be  accurately  determined  by  com- 
parison with  a  government  standard,  or  with  a  well  authenticated 

copy. 

*  In  \  I.  we  have  made  several  long  quotations  from  an  admirable  "Syllabus  of 
a  Course  of  Lectures  on  Physics,"  by  Prof.  Joseph  Henry;  for  it  was  not  in  our 
power  to  have  expressed  these  important  fundamental  truths  so  well,  and  we  do  not 
know  of  any  work  in  which  they  are  so  concisely  stated. 


12  Lecture- Notes  on  Physics. 

Two  units  of  length  are  used  in  this  country :  the  yard  (subdivided 
into  feet,  inches,  and  tenths  of  inches),  and  the  metre  (subdivided  into 
tenths,  hundredths,  thousandths,  and  so  on  decimally). 

The  yard,  originally  derived,  about  1120,  from  the  length  of  King 
Henry  the  First's  arm,  was  accurately  fixed  by  Captain  Kater,  in 
1818,  when  he  obtained  the  ratio  of  the  length  of  the  standard  Eng- 
lish yard  to  the  length  of  a  pendulum  beating  seconds  (i>  e*  making 
86,400  vibrations  in  one  mean  solar  day)  in  vacuo  at  the  level  of  the 
sea,  in  the  latitude  of  Greenwich.  He  found  that  the  length  of  such 
a  pendulum,  from  the  point  of  suspension  to  the  centre  of  oscillation, 
was  39-13929  inches  of  Bird's  standard  of  1760,  at  62°  Fahr. 

The  actual  standard  of  length  of  the  United  States  is  a  brass  scale 
of  82  inches  in  length,  prepared  for  the  U.  S.  Coast  Survey,  by 
Troughton  of  London;  meant  to  be  identical  with  the  English  Impe- 
rial Standard,  and  deposited  in  the  Office  of  Weights  and  Measures 
at  Washington.  The  temperature  at  which  it  is  a  standard  is  62° 
Fahr.,  and  the  yard  measure  is  the  length  between  the  27th  and  the 
63d  inches  of  the  scale,  (See  "  Report  on  the  Construction  and  Dis- 
tribution of  Weights  and  Measures,"  by  Dr.  A.  D.  Bache.  1857.) 

Two  copies  of  the  new  British  standard,  viz  :  a  bronze  standard, 
No.  11,  and  a  malleable  iron  standard,  No.  57,  have  been  presented 
by  the  British  Government  to  the  United  States.  A  series  of  care- 
ful  comparisons,  made  in  1856,  by  Mr.  Saxton,  under  the  direction 
of  Dr.  Bache,  of  the  British  bronze  standard,  No.  11,  with  the 
Troughton  scale  &f  82  inches,  showed  that  the  British  bronze  standard 
yard  is  shorter  than  the  American  yard  by  0*00087  inch.  So  that,  in 
very  exact  measures  with  the  yard-unit,  it  is  necessary  to  state 
whether  the  standard  is  of  England  or  of  the  United  States,  since 
10,000  American  feet  =  10,000-5803  English  feet. 

"  The  Metre  is  a  standard  bar  of  platina,  made  by  Lenoir  in  Paris, 
which  has  its  normal  length  at  the  temperature  of  zero  centigrade, 
or  the  freezing  point.  Its  length  is  intended  to  make  it  a  natural 
standard,  and  to  represent  the  ten  millionth  part  of  the  terrestrial  arc 
comprised  between  the  equator  and  the  pole,  or  of  a  quarter  of  the 
meridian.  The  length  of  this  arc,  given  by  the  measurement,  ordered 
for  the  purpose  by  the  Assemblee  Nationale,  of  the  arc  of  the  me- 
ridian between  Barcelona,  through  France  to  Dunkirk  (about  9J 
degrees),  combined  with  the  measurements  previously  made  in  Peru 
and  Lapland,  gave  for  the  distance  of  the  equator  from  the  pole 
5,130,740  toises.  The  standard  toise  was  made  in  1735,  in  Paris,  by 


Lecture-Notes  on  Physics.  13 


Langlois,  tinder  the  direction  of  Godin.  It  is  a  bar  of  iron  which  has  its 
standard  of  length  at  the  temperature  of  1  3  °  Reaumur.  It  is  known  as 
the  Toise  du  Purou,  because  it  was  used  by  the  French  academicians 
Bouguer  and  La  Condamine  in  their  measurement  of  an  arc  of  the 
meridian  of  Peru.  The  toise  of  Peru  =  2-13145  English  standard 
yards.  They  found  an  ellipticity  of  3^4,  and  the  length  of  the  metre 
44:3-29596  lines  of  the  toise  du  Perou,  assumed  to  be  443-296  lines, 
or  3  feet  11*296  lines.  This  last  quantity  was  declared  in  1799  to 
be  the  length  of  the  legal  metre,  and  vrai  et  definitif,  and  is  the  length 
of  Lenoir's  platina  standard.  Later  and  more  extensive  measure- 
ments in  various  parts  of  the  globe,  however,  seem  to  indicate  that 
this  quantity  is  somewhat  too  small.  The  latest  and  most  exact 
results  we  now  possess,  combined  and  computed  by  Bessel,  would 
make  the  quarter  of  the  meridian  10,000,856  metres,  and  the  metre 
=  443-29979  Paris  lines.  Schmidt's  computation  would  make  it 
443*29977  lines,  and  both  numbers  are  confirmed  by  Airy's  results. 
The  legal  metre  is  thus,  in  fact,  as  Dove  remarks,  a  legalized  part  of 
the  toise  du  Perou,  and  this  last  remains  the  primitive  standard.  But 
it  must  be  added  that  a  natural  standard,  in  the  absolute  sense  of  the 
word,  is  a  Utopian  one,  which  ever-changing  nature  never  will  give 
us.  The  metre  is,  for  all  practical  purposes,  what  it  was  intended 
to  be,  a  natural  standard  ;  though  it  must  be  confessed  that,  in  prac- 
tice, the  question  is  not  whether  and  how  far  a  standard  is  a  natural  or 
a  conventional  one,  but  how  readily  and  accurately  it  can  be  obtained, 
or  recovered  when  lost."  (See  A.  Guyot's  Tables,  Meteorological 
and  Physical,  p.  111.) 

Captain  Kater  in  1818  determined  with  great  care  the  value  of  the 
metre  at  a  temperature  of  32°  Fahr.,  in  inches  of  Shuckburgh's  copy 
(made  by  Troughton)  of  Bird's  standard  yard  at  62°  Fahr.,  and  found 
one  metre  at  32°  Fahr.;  =39-37079  inches  at  62°  Fahr  .=  39-36850535 
United  States  standard  inches  at  62°.  Therefore  the  quadrant  of  the 
French  meridian  contains  393,707,900  English  standard  inches. 

It  has  recently  been  shown  by  M.  Schubert  that  the  equator  of 
the  earth  is  not  a  great  circle,  but  an  ellipse  ;  having  its  major  axis 
=  41,851,800  feet,  and  its  minor  axis  =  41,  850,  007  feet,  giving  an 
ellipticity  of  55g<jths  From  this  it  necessarily  follows  that  differ- 
ent quadrants  of  the  earth  will  differ,  being  longer  when  passing 
near  the  major  axis  than  when  passing  in  the  neighborhood  of  the 
minor  axis.  Consequently,  the  metre  is  only  the  Toutnjssi^h  Part 
of  the  quadrant  passing  through  Dunkirk.  Sir  John  Herschel  com- 


14  •  Lecture- Notes  on  Phycics. 

putes  4008  feet  for  the  excess  of  the  true  quadrant  over  that  assumed 
as  the  basis  of  the  metrical  system ;  which  makes  the  French  stand- 
ard ^Jgth  of  an  inch  too  short. 

Sir  John  Herschel  proposes  for  the  British  standard  an  aliquot 
part  of  the  polar  axis  of  the  earth,  as  a  natural  unit,  and  shows  that 
by  increasing  the  existing  British  standard  yard  (and  with  it,  of 
course,  its  subdivisions)  by  exactly  the  y^ijth  part  of  its  present 
length,  the  polar  axis  will  contain  500, 500, 000  of  inches  each  y^^th 
longer  than  the  present  standard  inch.  Herschel  also  shows,  that  by 
adopting  this  new  value  of  a  foot,  that  one  cubic  foot  of  distilled 
water  at  62°  Fahr.  will  contain  1000  ounces  if  we  increase  the  present 
ounce  only  the  y^th  of  a  grain ;  and  thus  there  will  exist  a  simple 
relation  between  the  measures  of  length  and  of  weight.  (See  Familiar 
Lectures  on  Scientific  Subjects,  Article,  "  The  Yard,  Pendulum,  and 
Metre,"  by  Sir  John  Herschel.  A.  Strahan :  London  and  New  York, 
1866.) 

In  1866,  Congress  passed  an  act  directing  that  15  grammes  of  the 
metric  system  of  weights  shall  be  deemed,  and  be  taken  for  U.  S. 
postal  purposes,  as  the  equivalent  of  one-half  ounce  avoirdupois. 

The  new  U.  S.  five-cent  piece  is  made  2  centimetres  in  diameter 
and  weighs  5  grammes. 

See  plate  I.  for  the  French  measure  of  one  decimetre,  divided  into 
centimetres  and  millimetres ;  and  the  English  measure  of  four  inches 
divided  into  halves  and  tenths. 

(The  standards  of  one  yard  and  of  one  metre  exhibited.) 

Measurement  of  lines  of  considerable  length.  There  are  two  modes 
of  directly  measuring  lines  of  considerable  length.  The  first  consists 
in  placing  in  contact  the  ends  of  rods  having  an  exact  standard  length, 
and  thus  applying  these  units  to  the  line  whose  length  is  required. 
The  second  consists  in  bringing  the  beginning  of  one  unit  to  coincide 
with  the  end  of  another,  by  bisecting  a  fine  X  mark  near  the  end  of 
one  of  the  rods  by  the  cross-threads  in  a  microscope  attached  to  the 
end  of  the  other  rod.  The  distance  on  a  rod  between  the  cross-hairs 
in  the  microscope  at  one  end  and  the  X  mark  at  the  other  being  equal 
to  a  standard  length. 

With  the  precise  apparatus  of  the  U.  S.  Coast  Survey,  a  base  of 
ten  miles  can  be  measured  so  accurately  that  the  error  made  is  only 
about  ±  y^th  of  an  inch.  (See  U.  S.  Coast  Survey  Eeport  for  1854, 
p.  103,  et  seq.j  for  a  description  of  this  apparatus  for  measuring  "base- 
lines.) 


Jonrnul  TraiiMm  Institute . 


ONE  DECIMETRE 


O      .005     01  .02  .03  .04  .05  .06  .07  .08  J09  .1 


One  Cubic   Centimetre  of  Distilled  Water  at  4°  Cr=  1  Gramme 


FOUR    INCHES 


One   Cubic    Inch   of  Water   at   62°  P- 252.  456    Grains. 


JV-  IT  Cmissa  Lot  3*  icDoA-'y.'.  !V,3) 


Lecture-Notes  on  Pliysics.  15 

The  Vernier  is  a  device  for  subdividing  still  further  the  lowest 
divisions  of  a  scale  without  dividing  directly  those  divisions  by  equi- 
distant lines.  It  consists  of  a  sliding  scale  calle$  the  vernier,  which 
glides  along  the  length  of  the  main  scale,  and  has  on  it  divisions 
which  differ  from  those  of  the  main  scale  by  a  known  fraction  of 
the  smallest  subdivision  on  the  main  scale.  Thus,  suppose  we  have 
a  main  scale  of  inches  divided  into  tenths,  and  we  wish  to  divide  the 
tenths  of  inches  into  ten  parts  by  a  vernier,  thus  giving  us  the  hun- 
dredths  of  inches :  we  take  a  length  of  nine-tenths  from  the  main  scale, 
and  place  it  on  the  vernier,  and  we  divide  this  length  into  ten  equal  parts, 
which  necessarily  gives  divisions  on  the  vernier,  each  of  which  is  T  J^th 
of  an  inch  less  than  the  smallest  division,  of  tenths,  on  the  main  scale ; 
therefore,  by  seeing  where  a  line  of  the  vernier  corresponds  with  a 
line  on  the  main  scale,  the  next  following  lines  of  the  two  scales  are 
distant  from  each  other  T  J  ^th  of  an  inch ;  the  next  following  lines 
differ  yj-^ths,  and  so  on;  and  thus  we  can  subdivide  any  y^th  of 
an  inch  of  the  main  scale  into  hundredths  of  an  inch. 

In  general,  to  read  to  the  nib.  part  of  a  scale  division,  n  divisions 
of  the  vernier  must  equal  n  -f  1  orn  —  1  divisions  on  the  main  scale, 
according  as  these  run  in  opposite  or  in  similar  directions. 

Large  models  exhibited  of  vernier  applied  to  straight  line  and  to 
arcs  of  circles. 

Rule  for  finding  the  smallest  reading  by  means  of  the  vernier. 
Divide  the  value  of  the  smallest  divisions  on  the  main  scale  by  the  num- 
ber of  divisions  on  the  vernier. 

This  beautiful  and  very  useful  invention  is  due  to  Pierre  Yernier, 
of  France,  who  first  described  it  in  a  work  entitled  "La  construction, 
I7  usage  et  les  proprietes  du  cadran  nouveau."  Bruxelles,  1631. 

The  Cathetometer  (from  Gr.  KMttos  vertical  height,  and  pit poi/  a  meas- 
ure), consists  of  a  vertical  rod,  which  rotates  round  its  axis,  and  carries 
a  telescope,  provided  with  a  spirit-level,  whose  line  of  collimation  is 
at  right-angles  to  the  axis  of  the  rod.  The  rod  rests  on  a  base  fur- 
nished with  levelling  screws  and  spirit-levels,  so  that  it  can  be  brought 
into  a  truly  vertical  position.  The  telescope  is  attached  to  a  vernier 
plate,  which  slides  along  the  length  of  the  rod  which  is  divided  into 
fractions  of  the  metre  or  of  the  foot.  The  instrument  is  used  to 
measure  the  vertical  distance  between  two  points,  whether  on  the 
same  vertical  line  or  not.  It  is  of  constant  use  in  making  all  meas- 
ures of  vertical  distances. 

(Instrument  exhibited,  and  method  of  using  it  shown.) 


16  Lecture- Notes  on  Physics. 

This  instrument  was  invented  by  Dulong  and  Petit,  and  subse- 
quently improved  by  Pouillet. 

Micrometer  /Screw;. (from  Gr.  jutxpo?  small,  and  uttpov  a  measure)  con- 
sists of  a  screw  with  a  large  circular  head,  whose  circumference  is 
divided  into  a  certain  number  of  equal  parts.  Suppose  the  screw 
has  50  threads  to  the  inch,  and  that  its  circular  plate,  which  rotates 
with  it,  is  divided  into  200  parts;  then  if  the  screw  is  turned  a 
whole  revolution,  it  will  advance,  in  the  block  in  which  it  turns, 
g'oth  of  an  inch ;  but  if  the  head  is  revolved  only  through  2  J  Oth  of 
a  revolution,  the  screw  will  advance  the  y^^th  of  an  inch.  With 
a  good  micrometer  screw  we  can  measure  accurately  the  T  O's  Oth  of 
a  millimetre,  or  about  the  y^^^th  of  an  inch. 

(Large  model  exhibited,  also  the  instrument  itself  in  various 
forms.) 

Splierometer.  As  its  name  indicates,  it  is  used  to  measure  spheres ; 
or,  more  concisely,  to  determine  the  radius  of  any  spherical  surface ; 
as,  for  instance,  the  radius  of  the  surface  of  a  glass  lens.  It  was 
invented  by  the  optician  de  Laroue,  for  the  latter  determination. 
It  consists  in  a  micrometer  screw,  supported  vertically  by  a  tripod. 
The  points  of  the  feet  of  the  tripod  are  equally  distant  from  each 
other ;  and  they,  as  well  as  the  screw,  terminate  in  fine  nicely  rounded 
extremities.  The  point  of  the  screw  and  the  points  of  the  tripod 
are  brought  into  the  same  plane  by  placing  the  instrument  on  the  truly 
plane  surface  of  a  slightly  roughened  glass  plate,  and  bringing  the 
screw's  point  to  bear  on  the  plane  with  the  same  degree  of  pressure  as 
the  points  of  the  tripod.  This  is  attained  by  means  of  a  jointed  lever, 
which  moves  with  the  screw,  and  its  shorter  arm  bearing  on  the 
glass,  its  longer  arm  is  always  brought  into  the  same  position.  (This 
can  only  be  clearly  understood  by  examining  the  instrument.)  The 
instrument  is  now  removed  to  the  lens,  the  radius  of  whose  curva- 
ture we  desire,  and  the  point  of  the  screw  brought  down  upon  the 
glass  till  it  bears  with  the  same  degree  of  pressure  as  in  the  previous 
instance.  The  difference  of  readings  on  the  head  of  the  micrometer 
screw  gives  the  perpendicular  distance  of  the  point  of  the  screw  above 
the  plane  passing  through  the  points  of  the  feet  of  the  tripod ;  or,  in 
other  words,  the  height  of  a  segment  of  a  sphere,  having  for  radius 
the  radius  of  curvature  of  the  lens,  and  for  base  the  circumscribed 
circle  of  the  equilateral  triangle  made  by  the  points  of  the  three  feet 
of  the  spherometer. 

Calling  this  height  A,  the  radius  of  the  circle  passing  through  the 


Lecture-  Notes  on  Physics.  17 

feet  of  the  tripod  r,  and  the  radius  of  the  spherical  surface  of  the 
lens  K,  we  have 

2  R  —  h  :  r  :  :  r  :  h  ;     whence 

2  R/i—  h2  =  r>  therefore 


half  the  value  of  2  K  being  of  coarse  the  radius  of  curvature  of  the 
lens. 

The  spherometer  will  give  accurate  measures  to  the  TT)Vfit;h  of  a 
millimetre,  and  is  so  minute  in  its  measurements,  that  if  the  finger 
be  momentarily  placed  on  the  plate  of  glass  under  the  point  of  the 
screw,  and  we  again  bring  down  "the  screw  to  this  portion  of  the 
glass,  we  find  that  the  heat  from  the  finger  has  swelled  the  glass  at 
the  part  touched  into  a  protuberance  ! 

It  is  easy  to  see  how  this  instrument  is  also  used  for  measuring 
the  thickness  of  plates. 

Comparator.  Model  of  instrument  exhibited  and  described.  In- 
vented by  Lenoir  in  1800,  to  compare  standards  of  length. 

In  this  instrument,  as  well  as  in  the  following  one,  we  measure  a 
multiple  of  the  quantity  we  wish  to  estimate. 

Saxton's  Reflecting  Comparator  and  Pyrometer.  Model  of  this 
instrument  exhibited,  and  mode  of  using  it  shown. 

It  is  so  delicate  that  it  will  measure  with  precision  the  TaD^uu^ 
of  an  inch. 

"  This  very  ingenious  instrument  has  been  applied  to  comparing 
the  yard  of  the  kind  called  end-measure,  in  which  the  distance  be- 
tween the  two  ends  of  the  bar  is  the  length  of  the  standard  yard. 
One  end  of  the  bar  to  be  compared  abuts  against  a  fixed  support^ 
the  other  end  is  free  to  move,  the  whole  bar  being  supported  hori- 
zontally on  rollers.  The  free  end  presses  against  a  small  horizontal 
bar  or  slide,  connected  by  a  chain,  with  a  vertical  axis,  carrying  a 
small  mirror.  As  the  free  end  of  the  bar  moves,  the  slide  moves 
also,  and  turns  the  mirror.  A  scale  placed  horizontally  at  any  con- 
venient distance  from  the  bar  and  in  the  same  general  direction,  is 
reflected  in  the  mirror,  the  image  being  viewed  by  a  telescope  placed 
at  the  same  distance  as  the  scale,  and  directly  over  it.  The  distance 
of  the  reflected  image  of  the  scale  being  twice  that  of  the  scale  from 
the  mirror,  the  motion  of  the  mirror  is  shown  as  on  a  divided  circle 
of  that  radius,  and  the  smallest  movement  of  the  mirror  is  measurable." 

(See  "  Keport  by  Dr.  A.  D.  Bache,  to  the  Secretary  of  the  Trea- 
3 


18  Lecture-Notes  on  Physics. 

sury,  on  the  Construction  and  Distribution  of  Weights  and  Measures," 
Washington,  1857,  p.  15,  et  seq.) 

The  illustrious  Gauss,  of  Gottingen,  in  1827,  first  used  this  method 

of  reflection  in  his  measures  of  the  forces  of  terrestrial  magnetism. 

It  also  can  be  applied  to  the  galvanometer,  by  attaching  its  needles 

to  a  very  light  silvered  mirror ;  and  thus  the  smallest  motions  of  the 

needles  can  be  shown  to  the  largest  audience. 

Dividing  Engines.  To  divide  the  straight  line  into  fractional 
parts,  and  the  circumference  of  the  circle  into  degrees,  minutes,  and 
seconds,  machines  are  used  which,  when  made  with  care  and  used 
with  skill,  are  capable  of  working  with  great  precision. 

To  divide  the  straight  line.  The  simplest  apparatus  for  dividing 
the  straight  line  is  that  described  by  Bunsen  in  his  "Gasometry" 
(London,  1857).  The  rod,  tube,  &c.,  to  be  divided  is  placed  in  a 
line  with  a  standard  scale,  and  with  long,  rigid  beam -dividers,  hav- 
ing one  of  its  points  on  a  division  of  the  scale  we  describe,  with  the 
other  point  a  line  on  the  rod  &c.  to  be  divided;  we  then  move  the 
point  of  the  dividers  to  the  next  division  of  the  scale,  describe  with 
the  other  point  another  line  on  the  rod,  and  so  on. 

If  the  weight  of  the  dividers  is  partly  supported  by  a  cord,  pass- 
ing over  a  pulley  overhead,  and  attached  to  a  weight,  and  an  assist- 
ant, with  a  glass,  carefully  places  the  point  on  a  division  of  the  scale 
while  the  operator  describes  the  line,  this  method  will  give  an  accu- 
rate copy  of  the  scale  from  which  we  transfer  the  divisions. 

The  most  accurate  machine  for  dividing  the  straight  line  consists 
of  a  plate  placed  on  A  shaped  guides,  and  moved  in  the  direction  of 
its  length  by  a  screw  working  in  a  nut  attached  to  its  lower  side. 
The  rod  &c.  to  be  divided  is  clamped  to  the  upper  surface  of  the 
plate,  and  a  cutting  tool,  which  can  only  move  in  bne  and  the  same 
plane  perpendicular  to  the  length  of  the  plate,  cuts  the  divisions  on 
the  rod  after  each  rotation  of  the  screw.  If  the  screw  contains  ten 
threads  to  the  centimetre,  then  at  each  whole  revolution  of  the 
screw  the  plate  will  advance  one  millimetre ;  and  any  fraction  of  a 
millimetre  can  be  cut  by  rotating  the  screw  that  same  fraction  of  a 
whole  revolution. 

So  minute  is  the  accuracy  of  which  this  machine  is  susceptible, 
that  it  is  able  to  cut  75,000  equidistant  and  parallel  lines  in  the 
breadth  of  an  inch,  and  each  division  is  distinctly  visible  under  the 
highest  power  of  the  microscope.  A  still  greater  number,  probably 
over  100,000  lines,  can  be  cut  in  the  length  of  an  inch,  but  they  can- 


Lecture- Notes  on  Physics.  19 

not  be  discerned  even  by  the  highest  powers  of  the  best  microscopes, 
furnished  with  the  most  perfect  means  of  illumination.  In  this  case 
our  mechanical  exceeds  our  optical  power. 

A  graduated  series  of  these  lines,  increasing  in  the  number  to  the 
inch,  are  used  under  the  name  of  Nobert's  test  lines,  to  determine 
the  denning  power  of  the  objectives  of  microscopes. 

Another  example  of  fine  engine  work  is  given  in  "  Barton's  but- 
tons," which  are  gold  buttons  stamped  with  a  hard  steel  die  on  which 
Barton  cut  hexagonal  groups  of  fine  equidistant  lines  with  a  diamond. 
On  account  of  the  interference  produced  in  the  light  reflected  from 
the  surface  of  these  buttons,  they  flash  with  the  brilliancy  and  the 
colors  of  gems. 

From  the  above  description  of  the  performance  of  the  dividing 
engine,  it  is  easy  to  see  how,  having  fine  lines  cut  at  known  inter- 
vals apart  on  plates  of  glass,  and  placed  in  the  ocular  or  on  the 
stage  of  a  microscope,  we  have  the  means  of  measuring  distances 
which  surpass  in  minuteness  the  determinations  of  the  micrometer- 
screw. 

To  divide  the  circumference  of  the  circle.  Two  methods  are  used 
in  the  graduation  of  the  circumferences  of  the  circles  of  astronomi- 
cal and  other  instruments. 

In  the  first,  and  probably  the  most  accurate  method,  the  circle  to 
be  divided  is  centered  and  clamped  upon  a  large  accurately  gradu- 
ated circular  plate,  which  revolves  on  a  vertical  axis,  and  under  a 
cutting  tool  which  moves  only  in  one  and  the  same  vertical  plane 
passing  through  the  centre  of  the  circle.  The  divisions  on  the 
periphery  of  the  large  and  originally  divided  circle  are  successively 
bisected  by  the  cross-wires  in  the  ocular  of  a  fixed  microscope,  by 
revolving  the  plate  under  the  microscope  by  means  of  a  tangent 
screw ;  and  at  each  bisection  the  tool  cuts  a  line  in  the  border  of 
the  circle  to  be  graduated. 

In  the  other  method  a  tangent  screw  works  in  the  notched  edge 
of  a  circular  plate;  and  by  one  whole  revolution  of  the  screw  the 
plate  is  carried  round  10'.  After  each  revolution  the  tool  cuts  the 
division.  Thus  the  circle  is  divided  into  arcs  of  10'  and  into  smaller 
fractions  of  a  degree  if  desired. 

The  first  method  is  used  in  Germany ;  the  second,  invented  by 
Ramsden,  and  subsequently  improved  by  Troughton  &  Simms,  and 
by  Gam  bey,  of  Paris,  is  practiced  in  England  and  in  this  country- 
At  the  U.  S.  Coast  Survey  Office  in  "Washington,  can  be  seen  a  very 


20  Lecture-Notes  on  Physics. 

superior  dividing  engine,  made  by.  Troughton  &  Simms,  of  London. 

In  both  methods  the  circle  of  the  engine  has  originally  to  be 
divided  by  the  principle  of  bisection.  To  bisect  a  straight  line,  one 
point  of  a  beam-compass  is  placed  at  one  extremity  of  the  line,  while 
with  the  other  point  we  strike  a  fine  line  nearly  bisecting  the  line 
we  wish  to  divide.  From  the  other  extremity  of  the  line  an  arc  is 
Struck  with  the  same  radius,  and  the  minute  space  included  between 
the  lines  thus  formed  is  bisected  with  a  fine  needle-point  under  a  mi- 
croscope. A  scale  of  equal  parts  is  thus  formed  with  the  most  extreme 
care ;  and  having  attached  to  it  an  accurate  vernier,  we  take  in  the 
beam-compass  from  this  scale  a  length  which  exactly  equals  the 
chord  of  an  arc  of  85°  20'  of  the  circle  to  be  graduated,  and  with 
this  chord  we  strike  on  the  circle  an  arc  of  85°  20'.  By  continued 
bisections  this  arc  is  subdivided  into  equal  parts  of  5'  each.  The 
60°  division  can  be  obtained  by  a  chord  equal  to  the  radius  of  the 
circle,  and  this  arc  of  60°  bisected  gives  30°,  which,  added  to  pre- 
vious arc  of  60°,  gives  an  arc  of  90°. 

The  reader  is  referred  to  an  article  on  "  Graduation,"  by  the  cele- 
brated artist  Troughton,  in  the  Edinburgh  Encyclopaedia,  for  the 
best  account  extant  of  this,  the  most  delicate  and  difficult  problem 
of  mechanical  art ;  which  has  tested  to  the  utmost  the  combined 
skill  of  the  first  mechanicians,  and  of  the  ablest  astronomers  of 
modern  times. 

2.  Instruments  used  in  the  Measurement  of  Angles. 

The  circumference  of  the  circle  is  divided  into  360  parts,  one  of 
which  is  called  a  degree — the  unit  of  angle  measures.  The  degree 
is  divided  into  60  minutes ;  the  minutes  into  60  seconds ;  the  sub- 
divisions of  the  second  are  decimal.  The  sign  of  the  degree  is  (°) ; 
of  the  minute  (') ;  of  the  second  ("). 

Theodolite,  Meridian  Circle,  Sextant,  <#c.,  described  from  the  instru- 
ments, and  from  diagrams, 

The  Micrometer  consists  of  two  spider-lines,  stretched  parallel  to 
each  other,  on  two  frames,  each  moved  by  a  micrometer  screw,  so 
that  the  threads  can  be  made  to  approach  to,  or  recede  from,  each 
other.  The  spider-lines  are  exactly  in  the  focus  of  the  object  glass 
of  an  achromatic  telescope,  so  that  the  image  of  any  object  formed 
by  the  object-glass,  and  the  lines,  are  in  the  same  plane,  and  there- 
fore both  equally  distinct,  when  viewed  through  the  ocular  of  the 


Lee  lure- Notes  on  Physics.  21 

telescope.  The  value  of  one  turn  of  the  micrometer  screw  may  be 
estimated  in  angle  units,  by  causing  the  spider-lines  to  embrace  the 
vertical  diameter  of  the  sun,  when  it  is  on  or  near  the  meridian ; 
and  then,  by  obtaining  its  exact  diameter  from  the  Nautical  Al- 
manac, we  can,  after  allowance  made  on  our  measure  for  refraction, 
reduce  a  turn  of  the  screw  to  its  value  in  angle. 

The  invention  of  the  micrometer  is  due  to  William  Gascoigne, 
1640,  who  also  first  applied  the  telescope  (invented  in  1609,  by 
Galileo)  to  a  graduated  circle.  Gascoigne  was  killed,  at  the  early 
ao-e  of  23,  while  fighting  for  Charles  I.,  at  Marston-Moor,  2d  July, 
1644* 

The  Reading  Microscope  is  a  similar  contrivance  to  the  Microme- 
ter ;  the  difference  being,  that  the  micrometer  spider-lines,  which 
cross  (x)  each  other,  and  are  moved  by  one  screw,  are  placed  in  the 
focus  of  a  microscope,  instead  of  in  the  focus  of  a  telescope.  The 
Eeading  Microscope  is  placed  over  the  divisions  of  a  graduated  cir- 
cle, and  is  so  adjusted  that  five  turns  of  the  screw  will  carry  the  point 
of  bisection  of  the  cross  (X)  lines,  the  length  of  the  smallest  division 
on  the  circle,  which  generally  equals  5'  of  arc.  The  head  of  the 
screw  of  the  reading  microscope  is  divided  into  60  parts,  So  that  when 
the  screw  is  revolved  through  one  of  these  parts,  the  cross  (X)  lines 
are  carried  over  the  circle,  a  distance  equal  to  one  second  of  arc.  In 
those  circles  used  in  the  most  accurate  determinations  of  astronomy, 
the  microscopes  read  to  y1^  of  a  second  of  arc ;  and  this  minute  ac- 
curacy is  absolutely  required  in  many  problems  now  in  process  of 
solution  by  Astronomers — (e.  g.  the  parallax  of  the  fixed  stars). 

The  difficulty  of  measuring  with  accuracy  to  a  fraction  of  a  sec- 
ond, will  be  readily  appreciated,  when  we  know  that  one  second  of 
arc  occupies,  on  a  circle  of  6  feet  in  diameter,  a  length  of  only  '0001745 
of  an  inch. 

For  an  excellent  method  of  stretching  spider-lines  in  the  micro- 
meter, devised  by  Lieut.  M.  F.  Maury,  see  Washington  Astronomi- 
cal Observations,  1845. 

Reflecting  Goniometer.  Used  in  the  measurement  of  the  angles 
of  crystals.  Invented  by  Dr.  Wollaston,  in  1809.  u  This  instrument 
will  give  the  inclination  of  planes  of  crystals,  whose  area  is  less  than 
iuiuotn  Part  °f  a  square  inch,  to  less  than  one  minute  of  a  degree." 
(Ency.  Metrop.  Art.  Crystallography.) 

*  See  Letters  of  Scientific  Men  of  the  Seventeenth  Century,  Oxford,  1841,  vol.  i., 
p.  33,  et  seq.  Letters  XIX.  and  XX.,  Will.  Gascoigne  toOughtred,  Dec.  2,  1640. 


22  Lecture-Notes  on  Physics. 

Instrument  exhibited,  and  method  of  using  explained,  by  meas- 
uring the  angle  of  a  crystal. 

The  Spirit  Level  is  formed  of  a  slightly  curved  glass  tube,  nearly 
filled  with  ether.  A  bubble  of  air  occupies  the  highest  portion  of 
the  convex  side  of  the  tube. 

Spirit  levels  have  been  made  of  such  extreme  delicacy,  that  an 
inclination  of  one  second  of  arc,  in  the  plane  on  which  the  level 
rests,  would  cause  the  bubble  to  move  three  millimetres ;  the  curva- 
ture of  the  tube  of  the  level,  in  this  case,  was  619  metres.  M.  Biot 
used,  in  his  measurement  of  the  arc  of  the  meridian  passing  through 
Dunkirk  to  Formentera,  a  level  whose  bubble  moved  one  millime- 
tre for  an  inclination  of  1/A*79, 

By  the  aid  of  the  levels  of  the  best  mechanicians  we  can  safely 
estimate  tenths  of  seconds  of  arc. 

The  level  appears  to  have  been  known  to  the  ancient  astronomers 
of  India. 

3.  Instruments  used  in  the  Measurements  of  Volumes. 

Volumes  of  liquids  and  of  gases  are  measured  in  graduated  cyl- 
indrical tubes. 

A  tube  or  vessel  of  any  form  may  be  graduated  into  cubic  centi- 
metres, by  marking  the  levels  reached  by  successive  equal  portions 
•of  mercury  poured  into  the  tube  or  vessel ;  each  portion  being 
<equal  to  the  exact  weight  of  one  cubic  centimetre  of  mercury,  of 
the  temperature  at  which  the  mercury  is  when  the  weighing  is 
performed.  It  is  better,  however,  first  to  divide  the  tube  into  equal 
divisions  of  length,-  say  millimetres,  on  a  dividing  engine,  and  then, 
with  a  short  tube,  whose  aperture  is  closed  with  a  ground  glass  plate, 
<and  whose  capacity  is  exactly  one  cubic  centimetre,  to  measure  off 
sand  pour  into  the  tiibe,  cubic  centimetres  of  mercury.  After  each 
addition,  the  number  of  millimetres  to  which  the  mercury  rises  is 
noted  in  a  table,  and  thence  the  value  in  capacity  of  each  division 
of  the  tube  is  deduced. 

For  further  information  on  this  subject,  see  Bunsen's  Gasometry, 
London,  1857  ;  and  the  papers  of  Eegnault,  published  in  the  Trans- 
actions of  the  Institute  of  France. 

Volumes  of  solids,  as  well  as  of  liquids,  may  be  determined  by  the 
•estimation  of  their  weight,  at  a  known  temperature,  and  under  a 
•known  atmospheric  pressure.  In  this  method  we  have  also  to  know 


Lecture- Notes  on  Physics.  23 

the  weight  of  a  cubic  inch,  or  of  a  cubic  centimetre  of  the  solid  or 
liquid.  The  advantage  of  the  French  gramme  weight  is  shown  in 
this  determination  of  volumes,  by  weight;  for  the  specific  gravity 
of  a  body  is  equal  to  the  weight  of  one  cubic  centimetre  of  the 
substance,  in  grammes  ;  for  one  cubic  centimetre  of  water,  at  4°  C., 
weighs  one  gramme. 

4.  Instruments  used  to  Measure  Weights. 

Two  units  of  weight  are  used  in  this  country :  the  grain,  fixed 
by  the  weight  of  one  cubic  inch  of  distilled  water,  at  62°  F.  and 
30  inches  of  the  barometer,  one  of  such  cubic  inches  of  water  weigh- 
ing 252-456  grains.  The  other  is  the  French  unit,  the  gramme, 
which  is  derived  from  the  weight,  in  vacuo,  of  one  cubic  centimetre 
of  distilled  water,  at  4°  C.  See  Plate  1. 

The  Balance.  Saxton's  U.  S.  Standard  Balance  exhibited  and 
described.  Becker's  Analytical  Chemist's  Balance.  This  balance 
will  weigh  to  the  T'<jth  of  a  milligramme,  with  each  pan  carrying 
a  weight  of  50  grammes;  or,  in  other  words,  its  beam  will  turn  with 
sni/rttnyth  of  that  weight,  the  index  being  deflected  jd  of  a  division 
of  the  scale  from  the  position  of  equilibrium. 

See  a  paper  by  "W.  Crookes,  in  the  Chemical  News  for  19th  April, 
1867,  On  the  Correct  Adjustment  of  Chemical  Weights;  and  the  in- 
vestigations of  Prof.  W.  H.  Miller,  of  Cambridge,  in  the  R.  S.  Philo- 
sophical Transactions  for  1856. 

Method  of  Double  Weighing.  Invented  by  Borda  of  Paris.  Con- 
sists in  counterpoising  a  mass,  whose  weight  we  desire,  by  shot,  sand, 
thin  copper  foil,  or  pieces  of  fine  wire,  and  then  replacing  the  mass 
by  weights,  until  equilibrium  is  again  established.  The  weights 
are  thus  placed  in  exactly  the  same  conditions  as  the  mass,  and  there- 
fore this  process  eliminates  all  errors  arising  from  unequal  length 
of  the  arms  of  the  balance,  etc.  This  process  is  far  more  accurate 
than  single  weighing. 

The  weights  of  very  small  portions  of  matter,  such,  for  example,  as 
chemical  traces,  can  be  estimated  by  the  deflection  of  a  very  fine  fila- 
ment of  glass,  one  end  of  which  is  cemented  to  the  edge  of  a 
block,  and  the  other  end  has  hanging  to  it,  a  still  finer  filament, 
supporting  a  disk,  about  ^^ggth  inch  thick,  made  of  elder  pith,  on 
which  is  placed  the  substance,  whose  weight  we  determine  by  the 
amount  of  deflection  it  causes  in  the  glass  filament.  With  this  ap- 


24  Lecture-Notes  on  Physics. 

paratus  (the  invention  of  the  author),  can  be  estimated  the  deflection 
produced  by  the  y^^th  of  a  millimetre. 

See  Silliman's  Journal  of  Science,  vol.  xxv.,  page  39,  1858,  for  a 
description  of  this  instrument. 

5.  Instruments  used  in  the  Measurement  of  Time. 

Definition  of  time,  from  Laplace's  Systeme  du  Monde,  ch.  Ill : — 
"Time  is  for  us  the  impression  which  is  left  in  the  memory  by  a 
series  of  events,  whose  existence  we  are  certain  has  been  successive. 
Motion  can  serve  to  measure  it ;  for  a  body  not  being  able  to  be  in 
several  places  at  the  same  time,  can  only  go  from  one  point  to 
another,  by  passing  successively  through  all  the  intermediate  points. 
If  at  each  point  of  the  line  which  it  describes,  it  is  animated  with 
the  same  force,  its  motion  is  uniform,  and  the  parts  of  that  line  can 
measure  the  time  employed  to  run  over  them.  When  a  pendulum 
at  the  end  of  each  oscillation  is  found  in  exactly  the  same  circum- 
stances, the  durations  of  the  oscillations  are  the  same,  and  time  can 
be  measured  by  their  number.  We  can  also  employ  for  that  meas- 
ure, the  revolutions  of  the  celestial  sphere,  which  are  always  equal 
in  duration  ;  but  all  have  unanimously  agreed  to  use  for  that  object 
the  motion  of  the  sun,  whose  returns  to  the  meridian,  and  to  the 
same  equinox,  or  to  the  same  solstice,  form  the  days  and  the  years." 

The  duration  of  one  revolution  of  the  celestial  sphere  constitutes 
a  sidereal  day  •  while  the  time  of  one  mean  revolution  of  the  sun 
equals  a  mean  solar  day.  The  sidereal  day,  which  is  used  by  as- 
tronomers, is  3m.  56'5s.  less  than  the  solar  day. 

Both  days  are  divided  into  24  hours  ;  each  hour  into  60  minutes  ; 
and  each  minute  into  60  seconds  ;  the  seconds  are  sub-divided  deci- 
mally. The  hours  are  designated  by  the  sign  h ;  the  minutes  by 
m ;  the  seconds  by  s.  Both  days  are  divided  into  86,400  seconds. 

The  real  unit  of  time  must  be  the  sidereal  day,  being  the  duration 
of  one  revolution  of  the  earth  on  its  axis.  From  recorded  ancient 
eclipses,  it  has  been  computed  that  the  time  of  this  revolution  has 
not  altered  by  7-3  Wffusstn  °f  its  length,  from  B.  C.,  720.  From  the 
sidereal  day  is  readily  deduced  the  value  of  the  mean  solar  day. 

The  Astronomical  Clock  A  clock  consists  of  a  train  of  wheel- 
work  (moved  by  the  descent  of  a  weight)  which  maintains  and 
indicates  the  vibrations  of  a  seconds  pendulum.  The  Astronomical 
clock  is  made  with  every  refinement  of  workmanship ;  it  has  gen- 


Lecture- Notes  on  Physics.  25 

erally  a  "  dead-beat "  escapement,  and  has  always  a  pendulum  which 
is  so  compensated  that  a  change  of  temperature  does  not  alter  the 
length  from  the  point  of  suspension  to  the  centre  of  oscillation  of 
the  pendulum. 

The  Chronometer  is  a  large  watch,  very  accurately  constructed, 
having  a  "  chronometer  escapement "  and  a  carefully  compensated  bal- 
ance-wheel. It  beats  half-seconds. 

It  is  not  required  that  a  clock  or  chronometer  should  keep  the 
exact  time,  but  it  is  required  that  the  daily  rate  of  variation  from  the 
true  time  should  be  constant. 

The  pendulum  clock  was  invented  by  Huyghens,  of  Holland,  in 
1658  ;  and  in  the  same  year  Hooke  of  England  applied  the  spring- 
balance  to  time-keepers. 

Seconds  Stop-  Watch  exhibited  and  described. 

A  Uniformly  Revolving  Cylinder  was  first  proposed  by  Dr.  Thomas 
Young,  for  the  measurement  of  intervals  of  time.  With  this  appa- 
ratus we  can,  in  certain  cases,  measure  the  y^^th  of  a  second. 
"By  means  of  this  instrument  we  may  measure,  without  difficulty, 
the  frequency  of  the  vibrations  of  sounding  bodies,  by  connecting 
with  them  a  point,  which  will  describe  an  undulated  path  on  the  re- 
volving cylinder.  These  vibrations  may  also  serve,  in  a  very  simple 
manner,  for  the  measurement  of  the  minutest  intervals  of  time ;  for 
if  a  body,  of  which  the  vibrations  are  of  a  certain  degree  of  fre- 
quency, be  caused  to  vibrate  during  the  revolution  of  an  axis,  and 
to  mark  its  vibrations  on  a  roller,  the  traces  will  serve  as  a  correct 
index  of  the  time  occupied  by  any  part  of  a  revolution;  and  the 
motion  of  any  other  body  may  be  very  accurately  compared  with 
the  number  of  alternations  marked  in  the  same  time,  by  the  vibrating 
body."  From  Dr.  Young's  Lectures  on  Natural  Philosophy  and  the 
Mechanical  Arts  (a  work  to  be  thoroughly  studied  by  all  students 
of  physics),  2d  Edit.,  London,  1845,  page  147.  See,  also,  an  article 
on  "Apparatus  and  Experiments,"  devised  by  the  author,  in  the 
Franklin  Institute  Journal,  for  November,  1867. 

The  revolving  cylinder  has  recently  been  applied,  in  connection 
with  the  electric  clock,  to  the  registration  of  transit  observations. 
The  apparatus  described,  with  diagrams. 

Rater's  Method  of  determining  small  intervals  of  time,  by  weigh- 
ing the  mercury  which  flows  from  a  small  orifice,  under  a  constant 
pressure. 

4 


26  Lecture- Notes  on  Physics. 

Revolving  Mirror.  Invented  by  Prof.  J.  Wheatstone,  of  London, 
who,  in  1834,  measured  with  it  the  velocity  of  electricity.  Improved 
by  Arago,  and  proposed  by  .him  to  the  Institute  of  France,  in  De- 
cember, 1838,  to  determine  the  velocity  of  light.  In  1850,  Foucault 
of  Paris,  so  perfected  its  construction  and  arrangement,  that  he 
measured  with  it,  accurately,  the  TsWiiuuvth  of  one  second  of  time. 

The  principle  of  the  apparatus  can  be  rendered  clear  in  a  few 
words.  Suppose  a  ray  of  light,  entering  a  dark  room  by  a  fine  slit, 
falls  upon  a  rapidly  revolving  mirror.  This  ray  is  reflected  from 
the  revolving  mirror  to  a  fixed  mirror,  distant  from  it  say  20  feet. 
The  ray  reflected  from  the  fixed  mirror,  falls  again  upon  the  revolv- 
ing mirror.  Now,  suppose  that  in  the  time  the  light  takes  to  go 
from  the  revolving  mirror  to  the  fixed  mirror,  and  to  return  to  the 
revolving  mirror,  that  by  its  rotation,  the  face  of  the  revolving  mir- 
ror changes  its  position :  then  this  causes  a  deviation  in  the  ray 
reflected  on  its  return  to  the  revolving  mirror,  compared  with  its 
direction  when  it  left  the  revolving  mirror  to  go  to  the  fixed  mir- 
ror. Therefore,  knowing  the  time  it  takes  for  the  mirror  to  make  one 
revolution,  we  can  compute,  from  the  deviation  produced,  the  time 
required  by  the  light  to  go  over  double  the  distance  from  the  re- 
volving mirror  to  the  fixed  mirror.  This  distance,  in  Foucault's 
experiment  of  1850,  was  only  4  metres  (13*12  feet);  and  the  mirror, 
making  800  turns  in  a  second,  produced  a  deviation  in  the  return 
ray  of  -f^  millimetre. 

In  1862  Foucault  repeated  his  measurements  with  a  far  more  pre- 
cise instrument,  and  increased  the  distance  of  the  mirrors  from  4  to 
20  metres  (65  feet  7*4  inches).  The  result  of  his  measures  gives  a 
velocity  of  light  equal  to  185,177  miles  per  second,  differing  slightly 
from  the  measure  obtained  by  astronomic  observations,  when  we  use 
for  their  reduction  the  solar  parallax,  with  the  increase  in  amount 
which  it  appears  from  recent  observation  we  should  adopt. 

The  most  important  result  obtained  with  this  apparatus,  was  the 
fact  that  light  moves  slower  in  water  than  in  air,  the  velocities  in  the 
two  media  being  as  3  is  to  4.  In  short,  the  index  of  refraction  ex- 
presses the  ratio  existing  between  the  velocities  of  light  in  the  two 
media. 

"  An  admirable  experiment  in  physics,  in  which,  by  the  power  of 
intellect  and  manual  skill,  we  have  succeeded  not  only  in  rendering 
sensible,  but  even  measurable,  the  time  employed  by  light  to  run 
over  a  path  of  20  metres,  although  this  time  barely  equals  the 


Lecture  ^ Notes  on  Physics.  27 

jth  of  a  second!  and  which,  if  we  repeat  so  as  to  vary  its 
elements,  and  thus  make  evident  the  constant  causes  of  error  which 
affect  the  result,  appears  capable  of  giving  a  determination  of  the 
velocity  of  light  altogether  as  precise  as  that  which  is  deduced  from 
astronomical  phenomena !"  For  further  information  of  this  method, 
see  "Essay  on  the  Telocity  of  Light,"  by  M.  Delaunay  of  the  In- 
stitute of  France;  translated  for  the  Smithsonian  Institution  by  Prof. 
Alfred  M.  Mayer,  1864. 

Calculating  Engines. 

The  great  importance  of  such  machines  in  giving  us  faultless  loga- 
rithmic, astronomical,  nautical,  and  other  tables.  Impossible  to  cal- 
culate and  print  any  extensive  table  without  making  errors. 

For  a  description  of  the  Arithmetical  Machine  of  Pascal,  see  Oevres 
de  Pascal,  vol.  ii.,  page  368,  et.  seq.  Hachette,  Paris,  1860. 

For  information  in  reference  to  Charles  Babbage's  celebrated 
engines,  see  Edinburgh  Review,  1834;  Taylor's  Scientific  Me- 
moirs, vol.  iii.,  1843,  London,  in  which  is  a  description  by  General 
Menabrea,  of  Babbage's  Difference  Engine,  and  of  his  more  recent  and 
far  superior  Analytical  Engine,  translated,  with  extensive  notes,  by 
the  accomplished  Lady  Lovelace,  the  only  daughter  of  Lord  Byron. 

See  also,  Passages  from  the  Life  of  a  Philosopher,  by  Charles  Bab- 
bage,  London,  1867. 

Exhibit  the  "  Tables  calculated,  stereomoulded,  and  printed  by 
machinery,  London,  Longmans  &  Co."  1857.  The  engine  by  which 
these  tables  were  calculated  was  made  by  George  and  Edward 
Scheutz,  of  Stockholm,  and  is  now  the  property  of  the  Dudley 
Observatory,  at  Albany,  N.  Y. 

TABLES  FOE  THE  COMPARISON  OF  THE  FRENCH  AND  ENGLISH  SYSTEMS 
OF  MEASURES  AND  WEIGHTS. 

Measures  of  Length. 

Denomination  and  Value.  Equivalent  in  English  Standard. 

Myriametre 10,000  metres        6-2137  miles. 


Kilometre1 1,000 

Hectometre 100 

Decametre...  10 


METRE  .........................  1  me  re        39-37079 


Decimetre 

Centimetre 
Millimetre 


0-62137  mile  =  3,280  feet  10  inches, 
328  feet  1  inch. 
393-707  inches. 


3-937 
0-3937 

'°8937 


28 


Lecture-Notes  on  Physics. 


Measures  of  Surface. 
Denomination  and  Value.  Equivalent  in  English  Measure. 

Hectare 10,0^0  square  metres  2-471  acres. 

ARE 100       "  119-6      square  yards. 

Centare 1       "      metre  1,550          "       inches. 

Measures  of  Capacity. 

Denomination  and  Value.  Equivalent  in  English  (Wine)  Measure. 

Kilolitre  or  Stere 1000  litres  =   1  cubic  metre  26417    gallons. 

Hectolitre  10J  '       =TVU         "  26-417      " 

Decalitre  10  «      =  10  "    decimetres  2-6417    " 

LITRE 1  litre  =    1  cubic  decimetre  1-0567  quart. 

Decilitre &  '       =  T V         "  0-845  gill. 

Centilitre T47  '       =  lu  cub.  centim.  0-338  fluid  oz. 

Millilitre ^  '       =1"         "  0-27       "  drm. 

Weights. 

Denomination  and  Value.  Equivalents  in  English  Avd.  Weight. 

Millier  or  tonneau   =  l)0r,0,000grms.=    1  cub.  met.  of  W.@4°  C.  =  2204-6    Ibs. 

Quintal =--     100,OOJ      '      =    1  hectolitre       "  =   22046    « 

Myriagramme ==       10,000      '      =  10  litres    "         "  =     22-046  •< 

Kilogramme  or  kilo  ;-          1000      '      =    1  litre      "         «  =     2-2046" 

Hectogramme =  100      '      =    1  decilitre         "  =     3-5274  oz. 

Decagramme —  10      '      =  10  cubic  centim.  =     0-3527   " 

GRAMME =r  1     ««      =    1     «        "  =   15-43235  gs 

Decigramme =  TV     "      —  ^     "         "  =     1'5432     u 

Centigramme =  ^     "      =  10  cub.  millim.  =•     0-1543     u 

Milligramme =         ^o     "      =    l     "         "  "     '      =     °  °154    " 

Inch  =      25-39954      millimetres. 

Foot  =        3-047945    decimetres. 

Mile  =1609315  metres. 

Square  Inch  =       6-451367    square  centimetres. 
'<        Foot=        9-28997        "        decimetres. 
*'       Yard  =      8360971        « 
"        Acre=:          -4046711  of  a  hectare. 
"        Mile=    258-9895         hectares. 
Cubic  Inch   =      16-38618      cubic  centimetres. 

"      Foot    =     28-315312       "    decimetres,  or  litres. 
Gallon  =        4-54346      litres. 

"  =  277-274          cubic  inches. 

Grain  =      64  79896       milligrammes. 

Pound  Avd.  =   453-5927        grammes. 

§  III.  Methods  of  Precision. 

In  §  II  we  described  various  instruments  used  in  precise  meas- 
urements, and  we  will  now  explain  certain  methods  used,  both  in 
making  a  series  of  measures  on  a  quantity  and  in  subsequently  com- 
bining them,  so  that  the  final  result  has  the  greatest  possible  accu- 
racy. These  methods  are  designated  as  methods  of  precision.  They 
can  be  classed  under  three  heads : — 

1.  Method  of  means,  or  of  averages. 

2.  Method  of  multiplication. 

3.  Method  of  successive  corrections. 


Lecture- Notes  on  Physics.  29 

1.  Method  of  Means. — If  a  series  of  measurements  be  taken  with 
an  instrument  of  precision  on  one  and  the  same  quantity,  it  will  be 
found,  when  all  the  measures  are  made  in  the  same  circumstances 
and  with  equal  care,  that  they  will  differ  from  each  other  by  &  small 
amount. 

The  mean  value  of  all  the  measures  is  taken  when  the  probability 
is  even  that  each  measure  is  more  or  less  than  the  true  measure  by  a 
small  quantity.  The  mean  result  in  this  case  is  the  same  as  if  we 
added  the  mean  to  itself  as  often  as  there  are  separate  measures,  and 
divided  by  the  number  of  measures. 

The  determination  of  means  is  of  such  constant  occurrence  in 
physics,  astronomy  and  chemistry,  that  the  discussion  of  their  de- 
gree of  precision  is  very  important.  It  is  founded  on  the  principle 
of  the  theory  of  probabilities;  and  we  will  here  give  a  few  of  its 
more  important  results. 

Omitting  from  consideration  gross  errors,  which  can  always  be 
avoided  by  operating  with  care,  the  causes  of  the  errors  of  meas- 
urements can  be  classified  in  two  groups,  1.  Constant  or  regular 
causes;  2.  Irregular  or  accidental  causes;  whence  we  have  two 
kind  of  errors,  1.  Constant  or  regular  errors,  which  are  reproduced 
when  we  repeat  the  observations  in  the  same  circumstances ;  and 
2.  Irregular  or  accidental  errors  of  which  we  are  not  able  to  get  en- 
tirely rid;  but  which,  not  being  subject  to  any  law  connecting" them 
with  the  circumstances  of  the  measurements,  occur  indifferently  to 
increase  or  to  diminish  the  true  measure. 

Examples.  A  constant  cause  of  error  would  bean  error  in  the  length 
of  the  unit  used  in  measuring  the  base-line  of  a  trigonometric  survey. 
This  error  will  be  regular  and  constant,  and  being  made  every  time  the 
unit  is  applied  to  the  length  to  be  measured,  it  will  be  in  proportion 
to  this  length.  It  is  evident-  that  this  error  can  be  allowed  for,  when 
we  know  its  amount,  and  thus  our  result  is  the  same  as  if  we  ope- 
rated with  an  accurate  unit.  As  examples  of  irregular  sources  of 
error,  we  may  instance  the  measurement  or  sighting  of  angles  in  a 
survey  ;  and  the  bisection  of  a  line  by  the  reading  microscope,  or  by 
the  telescope  of  a-catheometer. 

It  is  evident  that  constant  errors  are  not  eliminated  by  increasing 
the  number  of  observations;  but  the  accidental  errors  tend  to  dis- 
appear from  the  mean  as  we  obtain  it  from  a  greater  number  of  meas- 
ures ;  for  this  class  of  errors  are  as  likely  to  be  in  one  direction  as 
in  another  in  successive  trials,  and  therefore  the  mean  of  an  infinite 


30  Lecture-Notes  on  Physics. 

number  of  measurements  on  the  same  quantity  would  give  the  great- 
est attainable  precision,  for  the  quantities  would  then  exactly  balance 
each  other  in  excess  and  defect. 

To  find  the  degree  of  precision  of  a  mean  we  divide  the  whole 
series  of  observations  into  two  or  three  groups,  selected  at  random, 
and  we  calculate  for  each  its  mean.  If  these  means  differ  very  little 
we  can  regard  the  observations  contained  in  each  group  as  being 
sufficiently  accurate.  To  proceed  thus,  we  must  have  a  considerable 
number  of  observations,  and  they  must  not  contain  very  discordant 
measures.  If  the  partial  means  do  not  thus  agree,  the  precision  of 
the  general  mean  is  very  doubtful. 

Example.  In  the  observation  on  the  temperature  of  Providence, 
K.  I.,  by  Prof.  A.  Caswell,  during  twenty  years,  the  mean  tempera- 
ture of  the  first  ten  years  is  48°'l  F. 

"     "     second  "      "      "  .         .  48°-3 

Difference, .         .  0°'2 

The  mean  of  the  whole  twenty  years  is        .         .         .         48°*2 

This  shows  that  one  decade  is  sufficient  to  give  the  mean  tempe- 
rature to  within  about  one-tenth  of  a  degree. 

It  often  happens  that  we  find  the  measures  grouped  together  in 
series,  depending  on  the  periods  or  on  the  various  circumstances 
when  they  were  made.  It  is  then  indispensable  to  calculate  the 
partial  means  of  these  series,  and  if  there  has  existed,  during  the 
time  of  the  observations,  a  constant  cause  of  errors,  but  of  variable 
intensity,  it  may  manifest  itself  by  taking,  as  above,  the  means  of 
several  groups. 

Errors  of  Observation,  called  also  residual  errors,  are  the  differences 
between  each  particular  measure,  and  the  mean  of  all  the  measures. 
These  errors  are  always  very  small  when  we  operate  with  care. 

The  absolute  value  of  the  quantity  sought  remaining  always  un- 
known, we  substitute  for  it  the  mean,  in  the  calculation  of  errors, 
without  which  it  would  be  impossible  to  appreciate  them. 

"When  the  errors  are  purely  accidental  it  is  found  that  they  can 
be  arranged  according  to  a  most  remarkable  law.  If  they  are 
grouped  by  their  signs  and  according  to  their  magnitude,  we  find 
that  the  positive  errors  and  the  negative  errors  are  equal  in  num- 
ber and  in  aggregate  value,  and  that  they  diminish  rapidly  as  we  go 
from  the  mean,  according  to  a  regular  law;  in  other  words,  the 
smallest  errors  are  the  most  frequent  and  are  principally  accumu- 


Jounutl  Franklin,  htstitutt 


\ 


X 


0 

THE      PROBABILITY     CURVE 


Lecture-Notes  on  Physics.  31 

lated  around  their  mean  value.  This  remarkable  fact  is  shown  in 
the  figure,  where,  on  the  left  of  the  axis  o  Y  are  placed  the  differ- 
ences or  errors  of  observation  less  than  the  mean,  or  '• — ;  and  on 
the  right  those  greater  than  the  mean,  or  -f . 

Theoretically,  this  curve,  called  the  curve  of  probability,  should 
be  perfectly  symmetric  and  regular.  If  it  departs  much  from  this 
form  it  shows  that  the  observations  or  experiments  have  not  been 
made  with  sufficient  care,  or  that  their  number  is  insufficient,  or 
that  there  exists  a  permanent  cause  of  error  arising  either  from  the 
observer  himself  (personal  equation),  or  from  the  manner  of  oper- 
ating. 

Suppose  we  have  now  found  a  series  of  measures  which,  being 
divided  into  groups,  give  as  many  means  which  differ  by  little ;  that 
the  difference  of  each  mean  from  the  general  mean  is  very  small ; 
and,  finally,  that  these  errors  are  distributed  about  the  mean  accord- 
ing to  the  curve  of  probability ;  then  the  cause  of  the  errors  is  purely 
accidental,  they  tend  to  diappear  from  the  mean,  and  the  theory  of 
probabilities  gives  us  the  numerical  value  of  the  degree  of  precision. 

The  first  principle  of  this  theory  is,  that  the  precision  in  the  mean 
increases  as  the  square  root  of  the  number  of  observations  from 
which  it  is  derived.  Thus,  the  circumstances  in  both  series  of  ob  • 
servations  remaining  the  same,  from  sixteen  observations  we  have 
in  favor  of  the  precision  of  the  mean,  a  probability  twice  as  great 
as  for  four  observations. 

"We  must  now  consider  another  element,  which  is,  the  error  of 
each  partial  measure,  or  its  departure  from  the  mean. 

We  call  the  mean  error  not  the  arithmetical  mean  of  the  errors, 
but  the  square  root  of  the  mean  of  the  squares  of  the  errors.  Let  *, 

*',  *",  &c., be  the  errors  of  observation,  their  mean 

error,  e,  will  be — 


e= 


n  being  the  number  of  observations. 

By  means  of  this  mean  error  we  can  calculate  what  is  called  the 
probable  error,  either  of  the  mean  or  of  a  single  observation. 

The  probable  error  is  that  quantity  which  has  such  a  magnitude 
that  there  is  the  same  probability  of  the  error,  in  the  quantity  de- 
termined, being  greater  as  there  is  of  its  being  less  by  this  quantity. 


32  Lecture- Notes  on  Physics. 

It  is  demonstrated  that  the  probable  error  of  a  single  observa- 
tion is — 

2 

r- 

The  probable  error  of  the  mean  result  is — 

E=- 


V 


n 


We  thus  see  that  it  diminishes  in  the  inverse  ratio  of  the  square 
root  of  the  number  of  observations. 

We  will  now  show  from  examples,  taken  from  the  appendix  to 
Gerhardt's  Traite  Elementaire  de  Ohimie,  how  the  foregoing  princi- 
ples and  formula  can  serve  to  indicate  the  exactness  of  a  series  of 
measures. 

M.  Dumas  made  experiments  to  determine  the  composition  of 
water,  and  to  verify  the  theoretical  law  of  Prout,  that  all  the  chemi- 
cal equivalents  are  exact  multiples  of  that  of  hydrogen.  The  equi- 
valent of  oxygen  being  represented  by  100,  that  of  hydrogen  should 
be  exactly  12'5.  But  a  series  of  nineteen  experiments  gave,  after 
the  corrections  were  made,  the  following  numbers,  which  we  arrange 
according  to  their  magnitude. 

12-472  12-508 

•480  -522 

•480  -533 

•489  -546 

•490  -547 

•490  -550 

•490  -550 

•491  -551 

•496  -551 

•562 

The  mean  of  the  numbers  is  12'515  ;  the  sum  of  the  squares  of 
the  errors  is  0'0173;  the  mean  error  e  =  0'030;  the  probable  error 
of  a  single  observation  is  0'020,  and  the  probable  error  of  the  mean 
result  E  =  0'0046.  It  would  appear  from  this  that  an  error  of  0'015 
is  impossible,  and  that  the  equivalent  of  hydrogen  ought  to  be  above 
12-500. 

But  if  we  observe  attentively  the  table,  we  will  see  that  the  num- 
bers, instead  of  being  accumulated,  according  to  the  law  of  proba- 
bility, around  the  mean,  tend  rather  towards  the  extremes.  The 


Lecture-Notes  on  Physics.  33 

means  of  the  two  columns  are  12486  and  12-542,  which  differ  too 
much  to  accept  the  hundredths  when  we  unite  them.  It  therefore 
follows  that  the  method  of  experiment  by  which  were  obtained  the 
numbers,  is  too  gross  to  decide  that  the  equivalent  of  hydrogen  departs 
from  12'5,  and  we  cannot  therefrom  decide  against  the  law  of  Prout. 

We  see  from  this  example,  taken  from  The  Theory  of  Chances, 
by  M.  Cournot,  how  important  it  is,  before  drawing  our  conclusions, 
to  examine  with  care  the  individual  observations  whence  we  deduce 
the  mean,  so  that  we  may  satisfy  ourselves  of  their  exactitude. 

We  often  exaggerate  the  accuracy  attained,  which  is  necessarily 
limited.  A  quantity  cannot  be  estimated  to  an  indefinite  precision 
even  when  the  observations  are  indefinitely  extended.  In  the  meas- 
ure of  a  length,  for  example,  whatever  care  we  may  take  we  can 
seldom  surely  rely  on  more  than  five  figures.  Thus,  one  of  the 
base-lines  in  the  triangulation  of  France  was  found  to  be  6075'9 
toises ;  we  find  that  there  can  easily  occur  in  this  number  an  error 
of  0*'!.  In  the  estimation  of  weights  there  is  also  a  limit  of  preci- 
sion which  we  cannot  surpass,  whatever  maybe  the  accuracy  of  the 
balance  and  the  skill  of  the  operator.  Here  also  we  cannot  rely  on 
more  than  five  significant  figures. 

Still  greater  reason  is  there  for  care  in  assuming  a  certain  preci- 
sion of  result  when  the  measure  is  the  result  of  several  different  physi- 
cal  operations,  each  of  which  brings  with  it  its  own,  error.  In  the 
determination  of  the  number  #  =  9m-809,  which  represents  the  inten- 
sity of  gravity  at  Paris,  (or  the  velocity  acquired  by  a  body  after 
falling  one  second,)  the  four  figures  which  we  have  given  are  those 
only  on  which  we  can  rely,  and  it  would  be  altogether  illusory  to 
extend  the  decimal.  If  we  compare  the  numbers  found  by  ablo  ob- 
servers, we  find  that  they  differ  even  in  this  fourth  figure. 

Intensity  of  gravity  at  Paris.  Length  of  second's  pendulum. 

Borda,  9'»-SOS9  Om-99385 

Biot,     9  -8091  -99387 

Bes*el,  9  -8094  -99390 

For  the  length  of  the  second's  pendulum  we  should  take  Om"9939, 
but  remembering  that  we  can  barely  rely  upon  the  result  to  11ffth  of 
a  millimetre. 

Physicists  and  chemists  have  often  to  estimate  the  exactness  it  is 
possible  to  attain  in  the  determination  of  densities  and  equivalents. 
By  repeating  these  operations  several  times  in  succession,  by  vary- 
ing the  methods,  by  taking  different  specimens  of  the  same  body, 
5 


34  Lecture  Notes  on  Physics. 

and  finally,  by  submitting  the  results  to  the  numerical  tests  which 
we  have  indicated,  we  arrive  at  the  degree  of  precision  of  the  results. 
We  thus  find  that  the  number  of  certain  figures  in  this  class  of 
determinations  is  less  than  we  would  d  priori  have  supposed,  and 
that  no  more  reliance  is  to  be  placed  in  several  of  the  decimals  that 
are  usually  retained,  than  if  we  wrote  them  down  at  random.  "We 
can  often  discover  this  influence  of  chance  in  a  series  of  measures. 
and  distinguish  those  decimal  figures  which  we  ought  to  suppress 
as  being  altogether  arbitrary,  since  they  are  less  than  the  errors  of 
observation.  Suppose  that  we  estimate  a  length  which  can  only  be 
appreciated  to  a  millimetre ;  if  we  retain  the  tenths  of  millimetre, 
they  will  be  entirely  of  an  arbitrary  value.  The  figures  written  in 
the  tenths  will  be  irregular  in  the  successive  measures,  absolutely, 
as  if  we  obtained  them  by  drawing  at  random,  out  of  our  pocket,  mar- 
bles having  pasted  on  them  the  figures  0, 1,  2,  3,  4,  5,  6,  7,  8,  9.  But,  as 
the  mean  of  these  ten  figures  is  4'5,  it  follows  that  if  we  have  a  great 
many  results,  the  mean  of  the  figures  of  tenths  of  millimetre  will  be 
exactly  4*5.  And  vice  versa,  if  in  a  long  series  of  numbers,  given  by 
observation,  we  find  that  the  mean  of  all  the  decimal  figures  of  the 
last  order  is  4*5,  we  can  omit  them.  If  it  is  the  same  with  the  fig- 
ures of  the  next  higher  order,  we  suppress  them  also  and  so  on. 

M.  Schrotter  determined  the  equivalent  of  phosphorus,  by  weigh- 
ing the  phosphoric  acid  produced  in  the  combustion  of  a  known 
weight  of  that  element.  These  experiments  were  made  upon  amor- 
phous phosphorus,  previously  dried  at  150°  C.,  in  an  atmosphere 
of  carbonic  acid  or  of  hydrogen.  The  transformation  of  the  phos- 
phorus into  phosphoric  acid,  was  effected  in  a  combustion  tube, 
through  which  passed  a  current  of  perfectly  dry  oxygen.  After 
the  combustion,  the  phosphoric  acid  was  sublimed  in  an  atmosphere 
of  oxygen,  so  as  to  oxydise  any  traces  of  phosphorus  acid  which 
might  have  been  produced. 


Phosphoric  acid  produced  by  one 
part  of  phosphorus. 


Lecture- Notes  on  Plnjsics.  35 

The  following  are  the  results : — 

2-28909 

-28783 
•29300 

•28831 
•29040 
-28788 
•28848 
•28856 
'28959 
•28872 

Mean,  ,  2-28919 

"We  find  that  the  sum  of  the  figures  of  the  last  column  on  the 
Tight  is  46,  the  mean  of  which  is  4*6,  since  there  are  ten  results.  In 
the  same  manner  the  mean  of  the  next  column,  to  the  left,  is  4-4.  We 
therefore  retain  only  the  first  three  decimal  figures,  the  remaining 
two  being  solely  due  to  hazard.  This  done,  if  we  calculate  the  error 
•of  each  observation,  or  the  departure  from  the  mean,  we  will  find 
that  the  sum  of  the  squares  of  these  errors  is  0'000022 ;  the  mean 
error  e  =  0'0015,  the  probable  error  of  a  skigle  observation  is  O'OOl, 
and  the  probable  error  of  the  mean  is  E  —  0'0003. 

We  can  therefore  admit  that  one  part  of  phosphorus  gives  2-289 
of  phosphoric  acid,  and  that  the  error  of  this  result  certainly  does 
cot  exceed  six  times  the  probable  error,  that  is  to  say,  0'002.  The 
-corresponding  equivalent  of  phosphorus  is  31*027,  and  its  probable 
^error  =  0'004,  which  is  expressed  thus — 

3 1-027  ±0-004. 

But  that  supposes  that  the  analyses  are  only  effected  with  accidental 
errors,  while  it  may  readily  happen  that  they  are  subject  to  a  con- 
.stant  error.  It  would  then  follow  that  the  figure  of  the  hundredths 
could  not  be  regarded  as  certain ;  therefore  M.  Schrotter  adopts  sim- 
ply 31  as  the  final  result  of  his  researches. 

It  is  from  the  constant  errors  existing  that  we  may  err  in  the 
conclusions  that  we  prematurely  draw  from  the  calculation  of  the 
mean  6f  the  probable  errors.  .And  it  often  happens  that  subse- 
quent observations  show  that  the  mean  adopted  at  first  is  affected 
with  an  error  far  exceeding  the  probable  error  which  we  attributed 
to  ^it. 


36  Lecture-Notes  on  Physics. 

It  is  a  characteristic  of  the  arithmetical  mean,  that  it  makes  the 
sum  of  the  squares  of  the  residual  errors  a  minimum. 

To  illustrate  'this  principle  geometrically,  suppose  that  several 
observations,  made  to  determine  the  position  in  space  of  a  point,  A, 
give  several  positions  around  the  true  point,  A.  Connect  the  points 
given  by  observation  by  straight  lines ;  then  the  centre  of  figure  or 
•  centre  of  gravity  of  the  polygon,  which  is  formed  by  joining  these 
points,  is  evidently  the  most  probable  position  of  A.  Draw  lines 
from  the  angles  of  the  polygon  to  its  centre  of  figure ;  then  these 
lines  will  represent  the  several  errors  of  the  observations.  Now,  it 
is  a  property  of  the  centre  of  gravity  of  such  a  polygon,  that  the  sum 
of  the  squares  of  these  lines  is  less  than  the  sum  of  the  squares  of 
similar  lines  drawn  to  any  other  point  within  the  polygon.  There- 
fore, if  the  sum  of  the  squares  of  the  errors  is  the  least  possible,  we  have 
the  nearest  mean  we  can  obtain  from  the  observations. 

This  is  the  fundamental  principle  of  the  celebrated  theorem  of 
Least  Squares,  of  Gauss  and  of  Legendre,  which  has  rendered  such 
inestimable  service  to  physics  and  astronomy. 

Suppose  that,  instead  of  having  to  determine,  as  in  the  above  ex- 
amples, a  single  unknown  quantity — which  is  the  mean  of  the  several 
observations, — we  propose  to  determine  several  unknown  quantities 
entering  as  functions  in  equations.  As  observations  or  experiments 
give  the  known  quantities  of  these  equations,  we  will  have  as 
many  equations  as  there  are  observations.  Having,  for  example, 
two  quantities  to  be  determined,  x  and  y,  suppose  that  we  have  made 
three  observations,  giving  three  equations.  The  values  of  x  and  y 
deduced  from  two  of  these  equations,  will  not  generally  satisfy  the 
third.  The  difficulty  consists  in  using  all  the  equations  at  once, 
so  that  we  can  deduce  from  them  a  series  of  values  which  will 
satisfy,  in  the  most  accurate  manner  possible,  all  the  equations.  The 
theory  of  the  method  of  least  squares  shows,  in  order  that  this  may 
happen,  that  the  sum  of  the  squares  of  the  errors  must  be  a  minimum. 

In  a  series  of  observations  or  experiments,  let  us  suppose  that 
the  errors  committed  are  denoted  by  e,  e',  efl ',  &c.,  and  suppose  that 
by  means  of  the  observations,  we  have  deduced  the  equations  of 
condition — 

ef    =  h'  -f  a!  x  -f  V  y  -f  cf  z  +  &c. 
e"  =  h"  +  a"  x  -f  b"  y  +  c"z  +  be.       [  (1.) 
<?"'  ==  h'"  +  a'"  x  +  bf"  y  +  c"f  z  +  be.  I 
&c.  &c.  &c. 


Lecture-Notes  on  Physics,  37 

Let  it  be  required  to  find  such  values  of  cc,  y,  z,  &c.,  .that  the  errors 
e,  e',  e",  e'",  &c.,  with  reference  to  all  the  observations  shall  be  the 
least  possible. 

If  we  square  both  members  of  each  equation,  in  group  (1),  and 
add  them  together,  member  to  member,  we  shall  have  — 


+  2  x     (ah+a'h'  +  &c.)  +  a  (&  ?/  +  c  2  +  &c.) 

4-  a!  (b'  y  +  c'  2?  4-  &c.)  -f  &c.    1  4-  &o., 

an  equation  which  may  be  written  — 

e2  j_  <.«     <5,/«      &(..=-  w=s  ^052      2     x  +  ^     &c. 


Now,  in  order  that  e2  +  ef2  -f-  &c.,  or  w,  may  be  a  minimum,  it  is 
necessary  that  its  partial  differential  co-efficients,  taken  with  re- 
spect to  each  variable,  in  succession,  should  be  separately  equal  to  0. 
Hence, 

^  =  o,  or  x  (a2 


=  =  2 

X 


-f  a  li  -f  u  hf  -f  &c.  +  a  (b  y  +  c  z  -f  &c.) 

-f  a'  (i7  y  +  c'  a  +  &o.)  +  &c.  =  0  ; 

and  similar  equations  for  each  of  the  other  variables.  Hence,  we 
deduce  the  principle,  that,  in  order  to  form  an  equation  of  condition 
for  the  minimum,  with  respect  to  one  of  the  unknown  quantities,  as 
x  for  example,  we  have  simply  to  multiply  the  second  members  of 
each  of  the  equations  of  condition  by  the  co-efficient  of  the  unknown 
quantity  in  that  equation,  then  take  the  sum  of  the  products,  and 
place  the  result  equal  to  0.  Proceed  in  this  manner  for  each  of  the 
unknown  quantities,  and  there  will  result  as  many  equations  as  there 
are  unknown  quantities,  from  which  the  required  values  of  the  un- 
known quantities  may  be  found  by  the  ordinary  rules  for  solving 
equations. 

Let  it  be  required,  for  example,  to  find  from  the  equations  — 

3  —    x  4-    y  —  %%=      ' 

5  —  3x  —  27/45*  = 
21__4<r—  y  —  ±z  = 
14  -f  x  —  By  —  3z= 

such  values  of  xt  y,  and  a,  as  will  most  nearly  satisfy  all  of  the 
equations. 

[It  is  to  be  remarked,  that  we  must  necessarily  admit  that  these 


38  Lecture- Notes  on  Physics. 

equations  already  approximate  closely  in  value,  that  is  to  say,  that 
the  values  of  x,  y,  and  2  deduced  from  the  three  first  will  satisfy 
approximately  the  third ;  without  this,  the  problem  is  absurd,  and 
the  observations  which  have  given  these  equations  are  unworthy  of 
confidence.] 

"Following  the  rule,  and  multiplying  the  first  member  of  each 
equation  by  the  co-efficient  of  x  in  that  equation,  we  get  the  pro- 
ducts— 

—  3  +       x —    y  +    22 
— 15+    9x  +  6y  — 152      • 

—  84  +  16z  +4y  +162 
14+      x  —  By—  32, 

and  placing  their  algebraic  sums  equal  to  0,  we  have — • 

27a+6y  — 88=0,       ....         (3.) 

Proceeding  in  like  manner  with  respect  to  the  unknown  quantities 
y  and  2,  we  obtain  the  equations — 

6cc  +  15?/  +  2  — 70=0,         .         .         .         (4.) 
y  + 54  s— 107  =  0,     .        .        .  (5.) 

Combining  equations  (3),  (4),  and  (5),  we  find — 

x  =  2-4702,  2/=3-5507,  and  2  =  1-9157, 

which  most  nearly  satisfy  all  of  the  equations  in  group  (2). 

To  show  the  practical  application  of  this  principle,  we  will  sup- 
pose that  it  is  required  to  investigate  the  values  of  a  constant  in  an 
equation,  by  means  of  several  independent  experiments. 

It  is  demonstrated  from  theory  that  the  length  of  the  pendulum 
which  beats  seconds,  in  any  latitude,  is  given  by  the  formula — 

L  =  x  +  y  sin.2  Z,       ....         (6,) 

in  which  L  denotes  the  length  of  the  pendulum ;  I  the  latitude  of  the 
place  on  the  surface  of  the  earth,  and  x  and  y  are  constants  to  be 
determined.  In  consequence  of  errors  incident  to  observation,  the 
values  of  x  and  y  cannot  be  accurately  determined  by  means  of  a 
single  observation ;  taking  the  metre  (denoted  by  ra)  as  the  unit  of 
measure,  suppose  that  the  length  of  the  second's  pendulum  has  been 
measured  in  six  different  places,  whose  latitudes  are  known,  and 
that  the  following  equations  have  been  deduced : 


Lecture-Notes  on  Physics.  39 

=  x  +  y  X  Om  .3903417  —  Om  .9929750 
=  x  +  y  X  Om  .4972122  —  Om  .9934620 
=  SB  +  y  X  Om  .5667721  —  0-  .9938784 
=x  +  y  X  Om  .4932370— Om  .9934740 
=  »  +  y  X  Om  .5136117  — Om  .9935967 
=  x  +  y  X0m. 6045628  —  Om , 9940932  ^ 

Applying  the  rule  already  deduced,  to  these  questions,  we  find 
the  equations — 

6  x  +  y  X  3m. 0657375  —  5m  . 9614793  =  0,         .        .        (8), 
x  X  m  .0657375  +  y  X  lm  .5933894  —  3™  .0461977  =  0,  (9.) 

Combining  equations  (8)  and  (9),  we  obtain — 
x  =  0m  .9908755,  y  =  0m  .0052942. 

Substituting  these  in  equation  (6),  it  becomes — • 

L  =  Om  .9908755  +  Om  .0052942  sin.2  Z,  .        .        (10.) 

By  means  of  formula  (10),  the  length  of  the  second's  pendulum 
may  be  found,  by  computation,  at  any  place  whose  latitude  is 
known.  In  like  manner,  the  method  of  least  squares  may  be  ap- 
plied to  a  multitude  of  similar  cases."  (See  art.  Least  /Squares,  Diet, 
of  Mathematics,  by  Davies  and  Peck,  N.  Y.,  1865.) 

It  is  impossible,  in  lectures  of  this  character,  to  give  more  than 
an  idea  of  this  fertile  principle  of  Least  Squares.  The  reader  is  re- 
ferred to  the  following  works  for  a  full  discussion  of  the  method. 
Methode  des  moindres  carrees  •  Memoir  es  sur  la  combinaison  des  obser- 
vations, par  Oh.  Fr.  Gauss,  traduit  par  J.  Bertrand,  Paris,  1855 ; 
also  Method  of  Least  Squares,  by  Prof.  William  Chauvenet,  in  the 
appendix  to  his  Spherical  and  Practical  Astronomy,  Phila.,  1863. 
From  this  work  the  figure  of  the  probability  curve  is  taken. 

2.  Method  of  Multiplication. 

The  method  of  multiplication  consists  in  measuring  a  known 
multiple,  by  n,  of  the  quantity  to  be  determined,  and  then  dividing 
the  result  by  n.  By  this  method  the  error,  committed  in  the  direct 
measure,  is  divided  by  n. 

Example  1.  To  find  the  diameter  of  a  fine  metallic  wire,  we  wrap 
a  certain  number  of  turns,  say  200,  of  the  wire  on  a  metal  cylinder, 
taking  care  to  press  the  coils  until  all  are  in  contact.  Measure  the 
length  of  the  200  coils,  and  dividing  this  length  by  200,  we  have — 
supposing  all  the  coils  are  formed  of  wire  of  the  same  diameter — the 


40  Lecture-Notes  on  Physics. 

diameter  of  the  wire  to  the  2^th  of  the  error  committed  in  the 
measure  of  the  length  of  the  200  coils.  Suppose2  for  example,  we 
have  made  an  error  of  ^  inch  in  measuring  the  length  of  200  coils; 
then  the  error  in  the  determination  of  the  diameter  of  the  wire  is 

2tjo  of  20  or  the  W<Jotn  of  an  incl1- 

Example  2.  By  a  similar  operation  we  determine,  with  great  ac- 
curacy, the  distance  between  the  contiguous  threads  (called  the  pitch) 
of  a  micrometer  screw. 

Example  3.  Saxton's  Beflecting  Comparator  is  a  beautiful  illus- 
tration of  the  principle  of  multiplication,  in  which  the  direct  mea- 
sure is  a  very  large  multiple  of  the  quantity  whose  value  we  desire. 

Example  4.  We  may  obtain  a  very  accurate  determination  of  the 
weight  of  a  body,  with  a  balance  with  equal  arms,  by  equilibrating 
the  body  with  shot,  &c.,  placed  in  the  other  pan ;  then  placing  the 
body  in  the  same  pan  with  the  shot,  &c.,  we  obtain  in  the  other  pan, 
in  shot,  twice  the  weight  of  the  body.  "We  now  add  the  shot  of  twice 
the  weight  of  the  body,  to  the  pan  containing  the  body,  and  on  again 
equilibrating,  we  have  in  the  pan  opposite  the  body,  a  weight  of 
shot  equal  to  four  times  the  weight  of  the  body,  and  so  on.  We 
finally  have  in  a  pan  a  known  multiple,  by  n,  of  the  weight  of  the 
body,  which, 'divided  by  n,  gives  the  weight  we  seek. 

Example  5.  The  Method  of  Repetition. — Describe  the  method  with 
the  aid  of  diagrams.  It  is  a  very  ingenious  application  of  the  me- 
thod of  multiplication  to  the  measurement  of  angles,  by  means  of 
the  repeating  circle,  invented  by  Tobias  Mayer,  of  Grottingen,  in  1762, 
and  subsequently  improved  by  Borda,  of  Paris. 

3.  Method  of  Successive  Corrections. 

This  method,  used  frequently  in  astronomy,  is  also  often  em- 
ployed in  researches  in  Physics.  We  will  explain  it  by  an  exam- 
ple (from  Daguin's  Traite  de  Physique,  vol.  i.,  page  30).  Let  it  be 
proposed  to  determine  the  capacity  of  a  ground  glass-stoppered 
flask.  If  we  knew  the  weight  of  the  water  at  4°  G.,  which  the 
flask  holds,  then  just  as  many  grammes  as  there  are  in  this 
weight,  are  there  cubic  centimetres  in  the  flask ;  for  a  gramme  is 
the  weight  of  one  cubic  centimetre  of  pure  water  at  4°  C.  It  is, 
therefore,  required  to  determine  the  weight  of  the  flask-full  of 
water  at  4°  C.,  which  we  suppose  is  the  temperature  when  the 
experiment  is  made.  We  first  weigh  the  flask,  full  of  air,  then 
full  of  water  at  4°  C.  Let  p  and  p  be  the  two  weights,  respectively. 


Lecture-  Notes  on  Physics.  41 

P  —  p  will  represent  the  weight  of  the  water,  if  the  flask,  during  the 
first  weighing,  had  contained  no  air,  whose  weight  is  added  to  that 
of  the  flask.  The  value  P  —  p  is  therefore  too  little  by  the  weight  of 
that  quantity  of  air.  Suppose  that  the  water  at  4°  C.,  weighs  n  times 
as  much  as  an  equal  volume  of  air,  taken  at  the  same  atmospheric 
pressure  and  temperature  as  in  the  experiment.  We  will  see  fur- 
ther on  how  this  number  n  is  obtained.  The  weight  of  the  air 

which  fills  the  flask  will  then  be  (p  —  p)  -,  if  we  suppose  for   the 

moment  that  P  —  p  is  the  exact  weight  of  the  water.  Then  the 
weight  of  the  water  which  fills  the  flask  will  be  — 

(p-p)  +  (r-p)  ^=(P-P)  (  1  +  1),         -.      .  .       (1.) 

This  expression,  however,  does  not  represent  the  exact  weight 
of  the  water,  because  the  term  (p  —  p),  and  consequently  the  term 

(p  —  p)  -,  are  too  small.     But  the  error  which  exists  in  value  (1) 

is  less  than  that  which  affects  the  value  P  —  p.  If  we  now  employ 
the  value  (1)  to  calculate  the  weight  of  the  air,  this  weight  will  be  — 

'  '" 


. 

and  the  weight  of  the  water  will  become  — 

/.  '  .    (2;) 


a  value  which  is  a  still  greater  approximation  to  the  true  value, 
and  which,  multiplied  by  1  :  n,  will  give  the  weight  of  the  air  with 
still  greater  precision. 

By  adding  this  weight  to  P  —  p,  we  will  have  for  the  weight  of 
the  water  — 


n     n 

nearer  than  any  of  the  preceding  values.     By  continuing  in  this 
manner,  we  can  carry  the  approximation  as  far  as  we  desire. 

§  IY.  Manners  of  Expressing  a  Law  —  Law  evolved  from  the  nume- 
rical results  of  Observations  and  Experiments. 

The  first  step  in  that  investigation,  which  has  for  its  object  the  dis- 
covery of  the  law  of  a  class  of  facts,  is  the  minute  examination  and  des- 
cription of  the  phenomena  and  of  the  circumstances  which  accompany 
6 


42  Lecture- Notes  on  Physics. 

them,  and  the  determination  of  those  conditions  necessary  for  their 
production.  This  having  been  accomplished,  we  observe  that  there 
always  exists  between  the  different  circumstances  of  an  associated 
class  of  facts,  relations  or  dependencies  which  bind  them  together 
in  such  a  manner,  that  if  we  change  one  of  the  circumstances  of  the 
phenomenon,  the  others  experience  determinate  modifications. 

For  example.  In  the  compression  of  air,  we  have  seen  that  the 
smaller  the  volume  into  which  the  air  is  forced,  the  greater  the 
force  of  compression  required ;  and  on  further  examination  with 
measurements  taken  of  (1)  the  volumes  occupied  (2)  under  different 
pressures,  we  find  that  the  relation  which  binds  (1)  and  (2)  is  that 
the  volumes  of  the  air  are  inversely  as  the  pressures  to  which  it  is 


Such  an  expression  of  the  relation  existing  between  the  different 
circumstances  of  a  class  of  associated  facts  is,  as  we  have  seen, 
(§  I.)  a  physical  law. 

11  These  laws  rule  all  phenomena,  and  are  their  most  complete 
representation.  Their  existence  did  not  escape  the  acute  minds  of 
the  philosophers  of  antiquity.  Plato, .  questioned  concerning  the 
occupations  of  the  Deity,  replied  that  He  geometrized  without  ceas- 
ing ;  wishing  thereby  to  express  according  to  Montucla,  that  the 
universe  is  governed  by  geometrical  laws"  (DAGUIN'S  TRAITE  DE 
PHYSIQUE,  Yol.  I.,  p.  8). 

If  the  law  is  the  expression  of  a  general  quantitative  relation 
existing  between  the  associated  facts  of  the  phenomenon,  it  can  be 
replaced  by  a  line  referred  to  coordinate  axes,  as  in  the  method  of 
Analytical  Geometry. 

Example  1.  Thus,  a  Parabola  represents  the  law  of  falling 
bodies,  because  the  origin  of  rectangular  coordinates  being  at  the 
principal  vertex,  the  abscissas  are  to  each  other  as  the  squares  of 
their  corresponding  ordinates ;  or, 

x'  :x"  '.:yf*'.y"*; 

and  the  law  of  falling  bodies  is  that  the  spaces  fallen  through  are  as 
the  squares  of  the  times ;  therefore,  if  any  ordinate  represent  in  units 
of  length  the  units  (seconds)  of  time  occupied  in  the  fall,  the  units 
of  length  contained  in  its  corresponding  abscissa  will  give  the  units 
of  length  fallen  through ;  one  unit  of  length  in  this  case — being  a 
function  of  the  intensity  of  gravity — is,  for  the  latitude  of  New 
York,  equal  to  16  feet  1  inch. 


MAT     JUN.    JUL.     AUG.      SEP    OCT.     NOV    DEC.     JAN.    FEB.    MAR.     APR 
1    16    1     15    1    16    1     15     1    14    1    14     1     13    1     13    1     16     1    15     1     17     1     15 


, 

, 

e 

/f 

t 

\ 

\ 

/ 

^ 

~M° 

A 

V 

1 

§ 

A 

/ 

\ 

/ 

\ 

/ 

•      ^ 

V 

/ 

\ 

00" 

/ 

\ 

g 

/ 

r 

/ 

\ 

\ 

\ 

\ 

1 

' 

I 

\ 

/ 

\ 

1 

\ 

I 

4 

\ 

% 

1 

1 

/ 

j 

/ 

V 

/ 

^^ 

fc  — 

^ 

2 

*>M« 

THERMAL    CURVE    OF    BALTIMORE    WITH      CORRESPONDING  SINUSOID. 


L1 
CO 


20       10       18      17      IB       15       14       13.     12       11        1O       9         8         7        fi         54         *        '2         L 


Lecture-  Notes  on  Physics.  43 

The  above  is  directly  shown  in  MORIN'S  MACHINE,  where  the 
body  describes  the  parabola  by  falling  parallel  to  the  axis  of  a  uni- 
formly revolving  cylinder,  against  which  a  pencil,  attached  to  the 
falling  body,  gently  presses.  The  curve  thus  formed  is  found  on 
spreading  out  the  paper,  which  covers  the  cylinder,  and  measuring 
its  abscissas  and  ordinates,  to  be  a  parabola.  Thus,  in  this  beauti- 
ful experiment,  the  body  itself  writes  on  the  paper  the  law  of  its 
motion. 

Example  2.  The  law  of  the  compression  of  gases  is  given  by  a 
curved  line  (see  Fig.  1,  Plate  III.  Curve  L  I/),  whose  abscissas  rep- 
resenting pressures  and  ordinates  volumes,  has  for  its  equation 


This  curve  is  an  equilateral  hyperbola,  and  therefore  the  axes  of 
X  and  of  Fare  asymptotes  to  the  curve  ;  and  this  should  be  so,  for 
the  gas,  supposed  to  be  nbn-liquinable,  has  under  an  infinitely  great 
pressure,  still  a  definite  volume,  which  is  expressed  by  the  ordinates 
corresponding  to  that  pressure.  If,  however,  the  gas  is  not  perma- 
nent (like  carbonic  acid),  the  curve  will  cut  the  axis  of  X  at  a  point 
corresponding  to  the  pressure  producing  liquifaction.  If  all  those 
gases  which  are  susceptible  of  liquifaction  —  though  the  pressures 
required  for  this  result  for  certain  gases  might  be  beyond  the  limit 
of  practicable  experiments  —  had  curves  whose  deflections  towards 
the  axis  of  X  all  followed  the  same  law,  we  could,  by  projecting 
the  curve  of  any  gas,  as  far  as  the  limit  of  experiment,  determine 
at  what  point  it  would  cut  the  axis  of  JT,  and  thus  determine  the 
pressure  of  liquifaction. 

See  Fig.  1,  Plate  III.  The  curve  L  I/  represents  Mariotte's  law, 
that  the  volumes  of  gases  are  the  reciprocals  of  the  pressures  ;  while 
L  C02  shows  the  fraction  of  a  unit  of  volume,  measured  on  the  axis 
o  Y,  occupied  by  carbonic  acid  gas  under  pressures,  from  1  up  to  20 
atmospheres,  measured  on  the  axis  o  x.  If  the  curve  of  hydrogen 
were  projected  on  this  diagram,  it  would  sweep  above  L  I/,  departing 
very  slightly  from  this  line  on  the  side  opposite  the  curve,  L  CO2. 

The  advantage  of  the  graphic  method  of  expressing  a  law  is,  that 
it  represents  to  the  eye  the  continuous  relation  between  the  associated 
facts  of  the  phenomenon,  and  therefore  gives,  by  direct  measurement, 
the  quantity  answering  to  any  measure  taken  of  either  of  the  related 
magnitudes  embraced  in  the  projected  curve.  Also,  by  the  mathe- 


44:  Lecture- Notes  on  Physics. 

matical  discussion  of  the  curve,  new  relations  which  otherwise  might 
pass  unknown,  may  be  evolved. 

Sometimes  the  relation  existing  between  the  associated  facts  is  too 
complicated  to  be  susceptible  of  a  concise  quantitative  statement. 
In  that  case,  we  lay  off  on  an  axis  of  rectangular  coordinates  lengths 
of  abscissas  equal  to  various  values  of  one  of  the  quantities,  and  on 
their  corresponding  ordinates  lengths  equal  to  the  corresponding 
values  of  the  other  quantity.  We  then  draw  tibera  manu,  through 
the  several  points  thus  determined,  a  curved  line,  and  this  line  will 
be  the  continuous  expression  of  the  relation  existing  between  the  two 
quantities  considered. 

It  may  happen  that  the  inspection  of  the  curve  leads  to  the  dis- 
covery of  a  law,  which  we  could  not  have  made  from  the  direct  com- 
parison of  the  numbers  given  by  experiment  or  observation.  For 
example,  if  the  curve  was  one  of  those  lines  studied  by  geometers, 
and  whose  properties  are  well  known,  the  relation  which  exists 
between  the  abscissas  and  ordinates  of  the  curve,  or,  in  other  words, 
its  equation  (referred  to  the  axes  X  and  ]T),  will  express  the  law 
which  we  seek.  If,  therefore,  we  recognize  by  mere  inspection,  that 
the  curve  resembles  a  known  line,  we  must  proceed  in  the  follow- 
ing manner  to  the  verification  of  that  supposition.  "We  write  the 
general  equation  of  the  curve,  referred  to  the  axes  X  and  Y,  giving 
it  indeterminate  co- efficients ;  we  then  successively  substitute  in  that 
equation  as  many  values  of  x  and  of  their  corresponding  ordinates 
y.,  as  there  are  co-efficients;  which  gives  equations  of  condition,  by 
means  of  which  we  calculate  the  values  of  these  co-efficients,  which 
we  then  substitute  in  the  general  equation.  We  then  successively 
substitute  for  x  and  y  other  numerical  values  given  by  the  experi- 
ments or  observations,  and  we  examine  if  these  numbers  satisfy  the 
equation.  If  this  is  the  case,  we  conclude  that  the  curve  is  really 
the  one  we  suspected,  and  its  equation  expresses  the  mathematical 
law  of  the  phenomena. 

For  a  gimple  example,  take  the  curve  described  by  Morin's  Machine 
for  showing  the  law  of  falling  bodies.  If,  from  preliminary  meas- 
urements, we  suspect  the  curve  to  be  a  parabola,  we  have,  for  the 
general  equation  of  that  curve — 


Lecture-Notes  on  Physics.  45 

in  which,  placing  xr  and  yr  from  measured  coordinates  of  the  curve, 
we  have  — 


substituting  this  value  of  the  parameter  in  the  general  equation,  and 
making  x  and  y  successively  x"  y"  ,  and  x"r,  y'n  ',  &c.,  obtained  from 
other  measurements  on  the  curve,  we  find  that  — 


whence  we  conclude  that  the  -curve  is  the  quadratic  parabola,  and 
therefore  — 

«,/  .  «,//  .  .  «./2  .  ,//»  . 

x    .  .x     .  .  y     .  y     j 

and  since  in  the  curve  described  by  this  machine,  x1  and  x"  are  equal 
to  the  spaces  fallen  through  by  the  body  in  the  corresponding  times 
yf  and  y",  we  have  the  law  that  the  spaces  fallen  through  are  as  the 
squares  of  the  times. 

Example,  in  which  more  complex  relations  exist  — 

"We  have  (in  a  paper  preparing  for  publication),  projected  the 
thermal  -curves,  of  the  variation  of  temperature  from  day  to  day 
throughout  the  year,  of  fifteen  places,  differing  in  latitude,  longi- 
tude, elevation  above  and  distance  from  the  sea,  and  in  other  topo- 
graphical features  ;  and  on  discussing  these  curves,  we  find  that 
they  can  all  be  replaced  by  sinusoids,  differing  from  each  other  only 
in  the  amplitude  of  their  flexures,  which  corresponds  to  half  the  annual 
range  of  temperature  of  the  different  stations. 

Lengths  of  abscissas  standing  for  days  and  lengths  of  ordinates  rep- 
resenting degrees  of  temperature  {see  Plate  III.,  Thermal  Curve  of 
Baltimore,  with  corresponding  sinusoid  ;  where  the  horizontal  red 
line  marks  the  mean  annual  temperature,  while  the  curved  red  line 
is  the  annual  thermal  variation,  and  the  curved  black  line  its  cor- 
responding sinusoid),  we  obtain  a  .series  of  points,  by  laying  off  on 
ordinates  erected  from  abscissas  corresponding  to  certain  days,  the 
mean  temperatures  of  those  days,  and  then  drawing  a  curved  line 
through  tKese  points,  we  have  the  annual  thermal  curve.  This 
curve  has  its  origin  at  the  point  of  the  axis  of  X  which  corres- 
ponds, in  the  average,  to  the  date  of  the  21st  of  April;  the  mean 
temperature  of  that  day  being  the  mean  temperature  of  the  year. 

(Diagrams  of  thermal  curves,  with  corresponding  sinusoids  of 
fifteen  stations  exhibited.) 


46  Lecture-Notes  on  Physics. 

Now  projecting  on  the  line  of  mean  annual  temperature  of  each 
curve,  as  the  base,  a  sinusoid  the  length  of  whose  base  (=360°) 
equals  the  length  of  the  year ;  whose  point  of  origin  is  on  the  base 
at  the  point  corresponding  to  the  21st  of  April,  and  whose  ampli- 
tude equals  the  half  yearly  range  of  temperature,  we  have,  in  every 
instance,  so  close  a  coincidence  between  the  thermal  curve  and  the 
sinusoid,  that  we  can  confidently  say,  that  the  variation  of  tempera- 
ture throughout  the  year  follows  the  variation  of  the  sinusoid  curve. 

The  equation  of  the  sinusoid  is — • 

y==sin.  x; 

and  as  the  sines  of  the  first  and  second  quadrants  are  -f  and  of  the 
third  and  fourth  — ,  it  follows  that  this  is  a  recurring  curve,  and  up 
to  sin.  (x  =  180°)  the  flexure  is  above  the  base  line,  while  from 
sin.  (x  =  180°)  up  to  sin.  (#  =  360°)  the  curve  lies  below  the  line, 
and  so  on;  each  succeeding  180°  being  on  the  opposite  side  of  the 
line  of  mean  temperature  from  the  preceding. 

Therefore,  expressing  any  date  in  days  reckoned  from  April  21st, 
and  converting  these  days  into  lengths  of  arc  —  x  at  the  rate  of  one 

onrvo  * 

day  =  ™,    =  0°  59'  I",  and  making  the  sin.  90°  or  E  =  £  the  mean 


yearly  range  of  temperature,  we  can  readily  determine  the  mean 
temperature  of  any  date  of  the  year. 

1.  Problem.    What  is  the  mean  temperature  of  the  10th  of  June, 
at  Paris  ? 

The  mean  annual  temp,  from  46  years  observations,  =  51'°  3  Fah. 
Half  the  yearly  range,         v.      ....     =15*°  47 
Days  from  21st  April  to  10th  June,      .      '  V  '     •"•    =50  days. 
The  above  number  of  days  in  arc,      V        .        .    =  49°  10'  50" 
Natural  sine  of  above  arc,   .     "   .        .        .-      V    ='756773 
Eadius  in  this  example,       .        .">.:*  :      .  ;-=15'47 
Then  the  temperature  of  15th  June  equals 

•756773  X  15°-47+51°-3        v'      .      "V       .    =-62°-90 
Mean  temperature  of  15th  June,  from  observa- 
tions of  46  years,    .       '.'.-'      .        ';""      .     =  62°-99 
Difference,         ?Jg3     .     =    0°'09 

2.  Problem.    What  is  the  mean  temperature  of  the  13th  of  Dec. 
at  Baltimore?     The  temperature  from  observations  =  35°'6.     Data 
for  solution.     From  35  years  observations,  the  mean  annual  tempe- 
rature »  54°-4.     The  yearly  range  —  44°-4, 


Lecture-Notes  on  Physics.  47 

3.  Problem.  Calculate  the  mean  temperature  at  Baltimore,  of 
of  the  14th  October.  The  observed  mean  temperature  of  this  day  is 
equal  to  56°'5. 

4>.  Problem.  Determine  which  is  the  warmest  and  which  is  the 
coldest  day  in  the  year  at  St.  Petersburg.  Data  for  the  solution. 
Mean  temperature,  38°*75.  Annual  range,  50°. 

From  observations  of  30  years,  the  warmest  day  at  St.  Petersburg 
is  on  the  average,  the  25th  July ;  the  coldest,  the  19th  January. 

We  can  therefore  determine  the  mean  temperature  of  any  day  of 
the  year  for  any  station  (not  situate  in  the  tropics),  whose  mean 
temperature  and  yearly  range  of  temperature  are  known.  The  mean 
temperature  of  any  place  can  be  found  by  taking  the  temperature 
once  a  month  during  one  year,  of  any  spring  which  issues  from  a 
depth  of  a  few  feet  below  the  surface  of  the  ground.  Thus,  the 
temperature  of  a  spring  in  Baltimore,  from  the  mean  of  one  obser- 
vation a  month,  during  one  year,  was  540<25,  only  0°*15  below  the 
mean  given  by  35  years  observations  of  the  thermometer  in  the  air. 

If  a  law  could  be  established  connecting  different  parallels  of  lati- 
tude with  the  yearly  range  of  temperature,  we  could  arrive  at  the 
other  datum,  and  then  the  curve  of  annual  variation  could  be  pro- 
jected from  a  knowledge  of  the  latitude  and  longitude,  with  the 
determination  of  the  mean  temperature  of  a  spring. 

This  method  of  representing  to  the  eye,  by  means  of  curved  lines, 
the  nature  of  the  laws  concealed  within  a  mass  of  numbers,  presents 
the  advantage  of  representing  the  continuous  and  periodical  nature 
of  cosmical  action.  It  also  presents  to  the  eye  a  form,  easy  to  seize 
and  remember,  instead  of  an  abstract  statement  referring  to  relations 
of  quantities,  difficult  to  be  conceived  and  sometimes  impossible  other- 
wise to  discover.  Thus  the  thermal  curve  of  Berlin  is  the  graphic 
expression  of  the  relation  of  the  365  days  of  the  year  to  observa- 
tions made  three  times  a  day  for  110  years,  or,  to  120,450  separate 
numerical  quantities. 

The  curves  show  by  their  deformities,  either  that  our  data  are  not 
the  results  of  sufficiently  extended  observations  to  give,  by  their  pro- 
jection, the  expression  of  a  law,  or,  that  another  cause  is  acting  which 
increases  or  diminishes  the  effect  due  to  the  cause  whose  rule  of  action 
is  expressed,  in  the  main,  by  the  curve. 

Thus,  in  the  thermal  curves  of  Greenwich,  of  Paris  and  of  Eome, 
an  upward  deflection  exists,  from  the  middle  of  January  to  the 
middle  of  April,  causing  a  departure  of  about  2°  Fahr.  from  the 


48  Lecture- Notes  on  Physics. 

sinusoidal  curves.  This  departure,  at  first  sight,  would  appear  as 
opposed  to  the  assumption  that  the  annual  variation  of  temperature 
follows  the  variation  of  the  sinusoid  ;  but,  in  reality,  it  is  "  an  excep- 
tion which  proves  the  law ;"  for  it  is  found  that  during  those  months 
the  prevailing  winds  for  those  stations  are  W.  S.  W.,  and  Dove  has 
conclusively  shown  that  in  the  northern  hemisphere,  the  thermometer 
stands  highest,  on  an  average,  with  a  S.  W.  wind.  This  deflection, 
however,  does  not  occur  in  the  curve  of  Berlin,  and  there  a  S.  17° 
W.  wind  prevals  during  those  months.  But  this  is  easily  accounted 
forr  and  the  explanation  will  at  the  same  time  give  a  sufficient  reason 
for  the  upward  deflection  of  2°  in  the  thermal  curves  of  Greenwich, 
Paris  and  Rome. 

The  prevailing  wind  of  Greenwich  from  the  middle  of  January 
to  the  middle  of  April,  reaches  that  station  after  having  traversed 
the  Atlantic  ocean,  whose  surface  water  has,  during  the  above 
months,  even  in  the  latitude  of  the  English  Channel,  an  average  tem- 
perature of  52°  Fahr.  Now,  one  cubic  foot  of  water  in  cooling  1° 
Fahr.  will  give  out  heat  sufficient  to  raise  3080  cubic  feet  of  air  1° 
Fahr.,  and  the  wind  after  traversing  this  surface  of  a  liquid,  of  high 
specific  heat,  13°*5  above  the  mean  temperature  of  the  air  during 
February,  passes  over  only  about  85  miles  of  lowlands  before  reach- 
ing Greenwich.  The  same  prevailing  wind  reaches  Paris  after  hav- 
ing also  passed  over  the  Atlantic, — of  even  a  slightly  higher  tem- 
perature than  in  the  case  of  the  Greenwich  wind, — and  traversed 
about  250  miles  of  land-surface  not  likely  to  affect  its  temperature, 
since  this  very  land  has  received  its  thermal  condition  mainly  from 
the  supernatant  air;  but  the  S.  17°  W.  wind  reaches  Berlin  after 
having  crossed  the  snow  covered  summits  of  the  Appenines  and 
Alps,  and  750  miles  of  land. 

This  general  graphic  method  of  expressing  the  relation  which  per- 
vades a  class  of  facts,  evidently  gives. a  ready  means  for  interpola- 
tion, and  even  where  deformities  exist,  a  mean  or  true  curve  can  be 
obtained  by  sweeping  a  line  through  the  mean  position  of  the  irregu- 
larly placed  points  given  by  the  observations  or  experiments.  By 
this  last  method,  Sir  John  Herschel  determined  the  orbits  of  several 
of  the  double  stars  (Transactions  of  the  Royal  Astronomical  Society, 
Vol.  Y.,  1832),  and  M.  Regnault  (Memoirs  de  TInstitut,  t.  21,  1847, 
page  316,  et  seq.,  and  page  574,  et  seq.),  from  his  measures  of  the 
tension  of  vapor  of  water  corresponding  to  different  temperatures, 
succeeded  in  ascertaining  the.  extreme  oscillations  of  the  errors 


Lecture- Notes  on  Physics.  49 

of  those  measures,  by  drawing  a  mean  curve  through  the  sphere  of 
the  points  given  by  his  experiments.  He  then  substituted  this 
curve  for  the  numbers  directly  determined  by  experiment,  because 
this  curve  expressed  the  relations  of  those  numbers  practically 
corrected  of  their  errors. 

It  should  be  remembered  that  each  number  at  which  we  finally 
arrive,  before  we  place  it  down  as  a  point  to  serve  in  the  projection  of 
the  curve,  is  the  most  probable  mean  which  we  can  obtain  from  the 
discussion  of  the  numbers  from  which  it  is  derived,  (see  §  III.),  and 
that  each  of  these  points  so  determined  is  arrived  at  independend- 
ently  of  any  other  point,  and  therefore  the  graphic  method  is  espe- 
cially adapted  to  combine  all  of  these  independent  determination  of 
points  in  one  curve,  and  to  make  them  mutually  correct  each  others 
errors. 

The  curve  thus  obtained  can  be  rendered  more  serviceable  in 
practice,  by  obtaining  from  it,  if  possible,  an  equation  which  will 
express  it ;  but  if  we  find  that  it  cannot  be  expressed  by  a  single 
equation,  we  can,  by  the  aid  of  formulae  of  interpolation,  find  the 
value  of  each  ordinate  corresponding  to  as  small  an  increase  in  each 
successive  abscissa  as  we  desire.  We  should  not,  however,  apply 
a  formula  of  interpolation  to  the  numbers  before  we  have  obtained 
from  them  a  mean  curve.  The  interpolation  is  merely  used  to  give 
a  ready  means  of  obtaining  the  values  of  any  related  magnitudes 
expressed  by  a  curve,  which  shows  the  relations  of  the  numbers 
corrected  of  their  errors ;  for,  as  M.  Eegnault  remarks,  "  the  graphic 
method  when  it  is  properly  executed,  is  preferable  to  all  methods 
of  interpolation  by  calculation ;  it  permits  us  to  distinguish,  imme- 
diately, the  variations  due  to  the  accidental  errors  of  the  observa- 
tions, and  the  constant  errors  which  depend  on  the  diversity  of  the 
methods  which  we  have  employed." 

§  Y.  The  General  Properties  of  Matter. — The  Constitution  of  Matter 
according  to  the  Molecular  Hypothesis. 

MATTER  is  that  which  affects  our  senses.     (See  §  I). 

A  body  is  a  portion  of  matter  limited  in  all  directions. 

In  the  different  manners  in  which  bodies  affect  our  senses,  con- 
sist the  properties  of  bodies.  These  properties  are  either  general, 
that  is,  belonging  to  all  bodies,  whether  solid,  liquid  or  gaseous, 
and  therefore  come  under  the  head  of  physics,  or  they  are  special, 
1 


50  Lecture-Notes  on  Physics. 

and  therefore  belong  to  the  province  of  chemistry.  (See  definitions 
of  Physics  and  Chemistry,  in  §  I). 

Matter  possesses  several  general  properties,  of  which  two  are 
called  essential  properties ;  for  without  them  we  cannot  conceive  of 
the  existence  of  matter,  and  therefore  they  can  serve  to  define  it. 
These  two  properties  are  extension  and  impenetrability. 

The  following  is  a  list  of  the  general  properties  of  bodies  : — 

1.  Extension.  ) 

2.  Impenetrability,  j     Necessar7  to  our  perception  of  matter. 

3.  Figure. 

4.  Divisibility. 

5.  Compressibility. 

6.  Dilatability. 

7.  Porosity. 

8.  Mobility.  ] 

9.  Inertia.  j     Ultimate  properties  according  to  the  mole- 

10.  Attraction  and  j  cular  hypothesis. 

11.  Eepulsion.       J 

12.  Polarity. 

13.  Elasticity. 

1.  Extension. 

Extension  is  the  property  which  every  body  possesses  of  occupy- 
ing a  portion  of  space,  which  we  call  its  volume.  A  body  always 
presents  the  three  dimensions — length,  breadth,  and  thickness ;  and 
it  is  by  abstraction  only  that  in  geometry  we  consider  surfaces,  which 
are  the  boundaries  of  bodies,  and  have  only  length  and  breadth; 
and  lines,  which  are  the  boundaries  of  surfaces,  and  have  only 
length;  and  points,  which  are  the  terminations  of  lines,  and  have 
alone  position. 

For  the  measurements  of  extension,  see  §  II. 

2.  Impenetrability. 

The  property  which  every  body  has  of  excluding  every  other 
body  from  the  space  which  it  occupies. 

In  ordinary  language,  we  say  that  one  body  is  penetrated  by 
another,  but  in  all  these  cases  it  is  found  that  the  particles  of  the 
one  body  are  merely  displaced  by  the  other. 

Examples.  A  needle  penetrating  (displacing)  mercury  contained 
in  a  fine  glass  tube.  The  mercury  rises  in  the  tube,  as  the  needle 


Lecture- Notes  on  Physics.  51 

descends,  to  an  amount  exactly  equal  to  what  would  be  produced 
by  pouring  into  the  tube  a  quantity- of  mercury  equal  to  the  volume 
of  that  portion  of  the  needle  below  the  surface  of  the  mercury. 

A  liquid  cannot  be  poured  into  a  vessel  unless  the  air,  which  it 
contains,  goes  out  as  it  enters. 

Consider  the  images  formed  in  the  foci  of  converging  lenses  and 
mirrors  ;  and  also  a  shadow. 

We  can  now  define  matter  as  all  that  which  has  both  extension 
and  impenetrability,  and  therefore  void  space  or  vacuum  is  space 
without  impenetrability, — being  penetrable. 

3.  Figure. 

The  bounding  surfaces  of  a  body  give  it  its  figure  or  form. 
That  department  of  science  which  discusses  and  classifies  the 
forms  of  nature,  is  called  Morphology. 

The  forms  of  bodies  may  be  arranged  under  two  divisions. 

I.  Regular  Forms.     II.  Irregular  or  Amorphous  Forms. 
Regular  forms  can  be  embraced  in  two  classes  (a  and  b). 

a.  Forms  of  matter  produced  by  the  action  of  forces  not  directed  by 
the  vital  principle. 

1.  Forms  of  the  heavenly  bodies.     Considered  in  Astronomy. 
Forms  of  liquids  in  motion;  as  "the  liquid  vein;"  (see  researches 

of  Poncelet,  Savart  and  Magnus),  the  rain  drop  ;  the  forms  of  waves. 

Forms  of  liquid  at  rest ;  (not  contained  in  vessels)  as  the  dew-drop  ; 
the  soap-bubble. 

Forms  of  liquids  not  subject  to  the  action  of  gravity ;  as  the  forms 
assumed  by  oil — suspended  in  a  mixture  of  water  and  alcohol  of 
the  same  density  as  itself — when  subjected  to  various  conditions  of 
molecular  action.  (See  Plateau's  experimental  and  theoretical  researches 
on  the  figures  of  equilibrium  of  a  liquid  mass  withdrawn  from  the  action 
of  gravity.  Translated  in  Smithsonian  Reports,  1863,  et  seq. 

2.  Forms  of  crystals.     The  laws  ruling  these  forms  considered 
in  the  science  of  Crystallography. 

b.  Forms  of  matter  produced  by  the  action  of  forces  directed  by.  the 
vital  principle. 

1.  Forms  of  plants.     Considered  in  botany. 

2.  Forms  of  animals.     Considered  in  zoology. 

All  the  forms  of  matter  not  contained  in  the  foregoing  classes  are 
irregular  or  amorphous,  and  belong  to  class  II. 

Diagrams  illustrating  above  classification  of  forms, 


52  Lecture-Notes  on  Physics. 

4.  Divisibility.    - 

Every  body  can  be  divided  into  many  parts,  and  these  parts  can 
be  further  subdivided  until  the  parts  become  so  minute  as  to  escape 
observation,  even  when  aided  by  the  most  powerful  microscopes. 

That  matter  was  susceptible  of  very  minute  division,  was  known 
to  the  philosophers  of  ancient  Greece  and  of  India,  and  they  dis- 
cussed the  -question,  whether  matter  was  infinitely  divisible,  or 
divisible  only  to  a  certain  minuteness,  when  the  parts  were  sup- 
posed to  be  unalterable  by  any  means,  whether  mechanical  or 
chemical.  Mathematically  speaking,  the  division  has  no  limit,  for 
however  small  the  residual  particles  may  be,  nevertheles's,  since 
they  have  extension  they  have  divisibility.  But  we  demand  not 
what  may  be  conceived,  but  what  really  exists  ;  and  in  fact,  the  dis- 
cussions of  the  ancients  are  about  the  infinite  divisibility  of  space, 
and  prove  nothing  as  to  the  actual  divisibility  of  matter. 

Anaxagoras  and  Aristotle  admitted  its  infinite  divisibility  ;  Leu- 
cippus  maintained  a  contrary  opinion,  and  Democritus,  siding  with 
Leucippus,  gave  the  name  of  atoms  to  the  ultimate  unalterable  parts 
of  which  he  supposed  all  bodies  to  be  composed.  Epicurus  attempted 
to  develope  the  ideas  of  Democritus,  that  found  in  the  seventeenth 
century  a  zealous  defender  in  Grassendi,  whilst  Descartes  upheld 
the  opinion  of  Aristotle. 

But  we  have  in  the  phenomena  of  molecular  physics,  in  the  laws 
of  combination  in  chemistry,  .and  in  the  facts  of  crystallography 
strong  evidence  that  all  bodies  are  composed  of  ultimate  excessively 
small  parts,  which  are  invisible,  even  with  the  aid  of  the  most  pow- 
erful microscopes,  but  still  of  finite  dimensions,  of  infinite  hardness, 
unalterable  by  any  means,  and  separated  from  each  other  by  spaces 
which  are  very  great  when  compared  with  their  size.  These  parts 
which  are  the  limits  of  the  possible  division  of  matter,  are  called 
atoms  (Gr.  d-ro/toj ;  »  privative  and  *e>v«,  to  cut.) 

According  to  this  hypothesis,  a  union  or  grouping  of  two  or  more 
atoms  forms  a  molecule;  a  combination  of  molecules  a  compound 
molecule,  and  a  union  of  the  latter  a  particle. 

The  sensible  division  of  matter  can  be  carried  to  a  very  minute 
limit. 

Examples.  Gold  can  be  beaten  into  leaves  ^^(jTjth  of  a  milli- 
metre thick. 

Silver  wire,  gilded  with  ?}vth  of  its  diameter  of  gold,  can  be 
drawn  so  fine  that  one  metre  weighs  only  eight  milligrammes.  The 


Lecture-  Notes  on  Physics.  53 


gold  film  on  this  wire  is  now  only  g^ss^  of  a  millimetre  thick. 
By  placing  a  short  piece  of  this  wire  in  nitric  acid,  the  silver  core 
is  dissolved  out,  leaving  a  tube  of  gold,  having  a  wall  only  the 
gWfiuoth  millimetre  thick.  Now,  under  the  best  microscopes,  we 
can  discern  a  surface  of  ^^th  °f  a  millimetre  in  diameter  ;  therefore 
we  can  divide  gold  into  particles  ?  ^^th  millimetre  in  diameter,  and 
BWfffifftn  millimetre  thick  ;  yet  each  of  these  parts  has  all  the  phy- 
sical and  chemical  properties  of  a  large  mass,  as  can  be  determined 
by  testing  it  under  a  microscope. 

Dr.  Wollaston  drew  platinum  wire  so  fine  that  its  diameter  was 
only  r  sVflth  millimetre  (3^5^  t>th  inch)  ;  and  although  platinum  is  the 
heaviest  of  the  metals,  yet  it  took  200  metres  of  this  wire  to  equal 
one  centigramme  in  weight  ;  or,  in  other  words,  one  mile  of  this 
wire  weighed  about  one  grain.  Dr.  Wollaston  accomplished  this 
by  wire-drawing  a  cylinder  of  silver  Jth  of  an  inch  in  diameter, 
having  in  its  axis  a  platinum  wire  ,-J^th  inch  diameter,  and  after 
having  obtained  a  very  fine  wire  —  having  in  its  interior  a  platinum 
wire  of  still  greater  tenuity  —  he  dissolved  with  nitric  acid  the  silver 
coating,  and  thus  obtained  an  almost  invisible  platinum  wire. 

The  thickness  of  a  soap-bubble,  in  the  dark  spot  which  is  formed 
on  it  just  before  it  bursts,  is  rWfftfstn  millimetre. 

Divisibility  of  matter  in  solution.  One  grain  of  carmine  tinges 
ten  pounds  of  water,  which,  we  can  divide  into  about  617,000  drops. 
If  we  suppose  that  100  particles  of  carmine  are  requisite  to  produce 
a  uniform  tint  in  each  drop,  it  follows  that  the  one  grain  of  carmine 
has  been  divided  into  62,000,000  parts. 

Metallic  solutions  and  chemical  tests. 

Illustrations  from  organized  bodies. 

The  thread  of  a  spider  is  composed  of  more  than  1000  separate 
threads. 

The  diameter  of  the  red  disks  contained  in  human  blood  is  ?^0  Oth 
inch  ;  while  the  diameter  of  the  blood-disks  of  the  Java  musk-deer  is 
only  the  r  »  1  25  incn,  so  that  a  drop  of  this  deer's  blood,  such  as  would 
adhere  to  the  point  of  a  fine  needle,  would  contain  150,000,000  disks. 

It  has  been  calculated  that  some  of  the  siliceous  plates  which  cover 
and  give  rigidity  to  the  minute  vegetable  cell-plants,  called  diato- 
maceae,  weigh  only  jfluoiuoffijti1  of  a  grain  ;  yet  the  surfaces  of  these 
plates  are  covered  with  the  most  exquisite  tracery  of  siliceous  stria 
or  bars,  often  not  more  than  ^g^^th  inch  in  width  and  thickness. 
Now  we  can  discern  a  surface  of  th  inch  in  the  best  micro- 


54  Lecture- Notes  on  Physics. 

scopes,  and  a  portion  of  one  of  these  siliceous  disks  of  that  area 
would  weigh  only  about  w^^^th  of  a  grain. 

Divisibility  of  odorous  substances. 

A  portion  of  musk  will  give  off  a  powerful  odor  during  a  year, 
and  yet  its  diminution  in  weight  has  not  been  sufficient  to  be  detected 
by  the  most  delicate  balance. 

"In  order  to  offer  an  inexact  idea  of  the  minuteness  of  the  par- 
ticles of  musk  which  are  still  capable  of  imparting  some  odor,  we 
state,  after  a  well  known  experimenting  physiologist,  that  a  certain 
liquid,  containing  as  much  of  an  extract  of  spirit  of  musk  as 
ssimffVfiffUfirti  part  of  its  whole  weight,  was  at  this  time  still  dis- 
tinctly odorous.  A  grain's  weight  of  a  liquid  of  which  s^^J^Tj^th 
part  was  of  that  extract,  spread  an  intensely  penetrating  odor.  Next 
after  musk  are  to  be  mentioned  certain  flower  ethers,  especially  the 
oil  of  roses,  a  little  drop  of  which  is  sufficient  to  fill  with  odor  an 
immense  atmosphere.  The  same  physiologist  states  that  a  certain 
space  filled  with  air,  of  which,  at  the  highest,  only  fTj^Jo"OTitn  Part 
was  vapor  of  oil  of  roses,  still  diffused  a  distinct  odor  of  roses." 
See  article  "  On  the  Senses,"  in  Smithsonian  Eeport,  1865. 

5.    Compressibility. 

All  bodies  when  subjected  to  exterior  pressure  are  reduced  in 
volume,  and  since  all  bodies  can  thus  be  indefinitely  compressed, 
it  follows  that  the  matter  of  which  bodies  are  composed  does  not 
fill  the  space  which  is  contained  within  their  bounding  surfaces ;  or, 
as  it  is  sometimes  stated,  a  body  does  not  form  a  plenum  of  matter. 

The  compressibility  of  solids  is  seen  in  all  structures  and  machines 
where  matter  experiences  great  pressure ;  e.  g.  The  stone  pillars 
which  support  the  dome  of  the  Pantheon,  at  Paris,  sensibly  dimin- 
ished in  height  as  the  weight  which  they  bear  was  placed  upon 
them,  and  a  visible  depression  takes  places  in  the  arches  of  massive 
bridges  when  the  "centres"  are  knocked  away. 

The  relief  of  coins  and  medals  is  produced  by  subjecting  disks 
of  metal,  placed  between  hard  steel  dies,  to  an  intense  and  sudden 
pressure,  which  is  so  great  that  the  metal  disk  is  not  only  changed  in 
form,  but  its  volume  after  being  struck  is  appreciably  less  than  before. 

For  a  long  time,  liquids  were  regarded  as  incompressible,  and 
this  opinion  seemed  to  be  confirmed  by  many  experiments  which 
the  Academy  of  Florence  made  during  the  end  of  the  seventeenth 
century.  Their  most  celebrated  experiment,  which,  however,  had 


Lecture-Notes  on  Physics.  55 

previously  been  made  by  Lord  Bacon,  consisted  in  filling  a  hollow 
silver  sphere  with  water,  and  after  stopping  the  orifice  with  a  screw- 
plug,  subjecting  it  to  the  powerful  pressure  of  a  screw-press.  It  is 
evident  that  if  the  sphere  is  flattened,  its  capacity  is  diminished, 
and  the  water  compressed.  They  were  surprised,  however,  to  see 
the  water  appear  on  the  exterior  surface,  as  a  fine  dew,  and  they 
thence  concluded  that  water  was  incompressible,  and  silver  porous 
to  this  liquid. 

John  Canton,  in  Trans.  E.  S.  1762,  first  showed  that  liquids  were 
compressible,  even  with  the  pressure  of  one  atmosphere.  His  appa- 
ratus consisted  of  a  large  thermometer  which  contained  the  liquid, 
and  the  bulb  of  which  was  inclosed  in  an  exhausted  receiver.  The 
liquid  was  thus  entirely  relieved  of  atmospheric  pressure.  The 
height  of  the  liquid  was  now  marked  upon  the  stem,  and  the  end 
of  the  sealed  thermometer  tube  being  broken,  the  air  was  allowed 
to  enter  the  stem  and  press  upon  the  surface  of  the  liquid,  and  to 
enter  the  receiver  and  press  upon  the  outside  of  the  bulb.  The 
liquid  instantly  fell  in  the  tube,  thus  clearly  showing  a  compression 
produced  by  the  pressure  of  one  atmosphere. 

Canton  thus  found  that  water  was  compressed  nj^Vuiiu^8  °f  its 
volume  by  the  pressure  ef  one  atmosphere,  and  this  determination 
is  quite  exact. 

In  1826,  Jacob  Perkins  described  in  the  Trans.  R.  S.  an  apppa- 
ratus  which  renders  the  compressibility  of  liquids  very  evident. 
Into  a  hollow  cylinder  passed  a  rod  through  a  stuffing-box,  and 
around  this  rod  tightly  fitted  a  leather  washer,  which  rested  on  the 
top  of  the  cylinder.  This  apparatus  being  filled  with  water,  or 
other  liquid,  was  placed  in  a  closed  cannon,  into  which  water  was 
forced  by  a  strong  pump,  until  a  certain  pressure  was  attained, 
which  was  measured  by  the  lifting  of  a  safety-valve  and  the  escape 
of  water. 

On  taking  out  the  cylinder  from  the  cannon,  the  leather  washer 
was  found  on  the  rod  several  inches  above  the  place  it  occupied 
before  the  experiment,  thus  showing  that  the  rod  had  entered  by 
that  quantity  into  the  vessel,  as  the  pressure  which  it  exerted  com- 
pressed the  liquid. 

Describe  and  use  (Ersted's  piezometer,  and  exhibit  table  of  Reg- 
nault  and  Grassi's  results  on  the  compressibility  of  liquids. 

Gases,  of  all  bodies,  are  the  most  easily  compressed.  This  fact  is 
readily  shown  by  enclosing  a  gas  in  a  cylinder,  into  which  glides 


56  Lecture-Notes  on  Physics. 

an  air-tight  piston.  By  pressing  on  the  piston,  it  is  forced  into  the 
cylinder,  and  the  gas  is  reduced  in  volume.  On  relieving  it  of 
pressure,  the  gas  expands  to  the  volume  it  had  before  the  experi- 
ment, and  will  do  so  no  matter  into  how  small  a  volume  it  may 
have  been  compressed,  or  how  often  the  experiment  may  have  been 
made.  In  this  respect  (of  perfect  elasticity)  gases  and  liquids  differ 
from  solids.  This  is  explained  by  the  fact,  that  the  ultimate  parts 
of  solids  have  polarity. 

It  is  found  that  by  pressure,  gases  are  reduced  to  volumes,  which 
are  the  reciprocals  of  the  pressures. 

6.  Dilatability. 

When  any  body  is  subjected  to  exterior  pressure  on  all  its  sur- 
faces, it  is  forced  into  a  smaller  volume,  but  on  the  pressure  being 
removed,  it  expands  to  the  dimension  it  had  before  the  pressure 
was  applied.  Also,  the  volumes  of  all  bodies  are  increased  by  a 
rise  in  their  temperature;  and  Cagnard — Latour  has  shown  that 
wires  and  rods  when  stretched,  have  their  volumes  increased. 

This  general  property  of  increase  in  volume  from  these  causes  is 
called  dilat ability. 

The  dilation  of  solids  and  of  liquids  is  shown  by  their  expan- 
sion, when  relieved  of  hydrostatic  and  atmospheric  pressures ;  and 
any  gas  expands  when  the  atmospheric  pressure  is  removed. 

Experiments.  The  expansion  of  solids  shown  when  relieved  of 
powerful  pressure.  The  experiment  of  Canton  reversed,  shows  the 
expansion  of  a  liquid  as  the  pressure  of  the  atmosphere  is  removed. 

The  expansion  by  heat,  shown  by  Saxton's  pyrometer,  of  liquids 
and  of  gases  by  large  liquid  and  air  thermometers. 

Experiments  of  stretching  wires  and  India-rubber,  with  Cagnard 
— Latour's  apparatus. 

7.  Porosity. 

The  spaces  which  exist  between  the  ultimate  parts  of  bodies — 
that  is,  between  their  atoms  and  between  their  molecules — are 
called  pores ;  by  virtue  of  which  all  bodies  have  porosity. 

This  property  is  a  natural  deduction  from  the  properties  of  com- 
pressibility and  dilatability ;  for  all  bodies  can  be  reduced  in  vol- 
ume by  pressure,  and  this  reduction  of  volume  is  only  limited  by 
the  pressures  which  our  machines  can  produce,  and  all  bodies  expand 


Lecture-Notes  on  Physics.  57 

in  volume  when  relieved  of  pressure ;  we  therefore  conclude  that  the 
ultimate  parts  of  bodies  do  not  touch,  but  are  separated  by  intervals 
that  we  call  pores. 

This  property  of  porosity,  first  indicated  and  concisely  defined  by 
Gassendi,  can,  however,  be  directly  proved  by  numerous  experiments. 

Porosity  of  solids.  Water  and  oil  have,  by  enormous  pressures, 
been  forced  through  the  walls  of  vessels  of  gold  and  of  iron;  iron 
converted  into  steel  by  absorbing  carbon  at  a  high  temperature ; 
gold  and  the  densest  bodies  are  transparent  when  reduced  to  suffi- 
ciently thin  leaves  ;  lead  is  so  porous  to  mercury,  that  Prof.  Henry 
made  a  syphon  of  a  solid  bar  of  lead  draw  mercury  over  the  side 
of  a  vessel  containing  it ;  marble  and  all  rocks  are  porous  to  gases, 
allowing  them  to  "diffuse"  through  them.  We  can  ascertain  how 
a  stone  will  be  affected  by  frost,  by  allowing  it  first  to  absorb  a  solu- 
tion of  sulphate  of  soda,  and  then  ascertaining  the  effects  produced 
by  its  subsequent  crystallization.  It  is  better,  however,  to  deter- 
mine directly  the  effect  of  the  freezing  of  absorbed  water.  See 
Smithsonian  Report,  1857,  p.  305.  (It  is  probable  that  the  two  pre- 
ceding instances  are  examples  of  the  porosity  existing  between  the 
integral  crystals  of  the  stone  and  not  between  the  ultimate  parts, 
and  therefore  is  not  porosity  in  the  sense  in  which  we  have  defined 
it.)  Hydrophane,  a  species  of  agate,  becomes  translucent  by  the 
absorption  of  water,  when  immersed  in  that  liquid,  while  the  dis- 
placed air  issues  from  the  hydrophane  in  minute  bubbles. 

M.  M.  Deville  and  Troost  have  recently  shown  that  platinum  and 
other  dense  metals  and  cast  iron,  when  heated  to  redness,  are  porous 
to  gases.  The  injury  to  health  produced  by  heating  dwellings  by 
means  of  cast  iron  stoves,  is  caused  by  the  vitiation  of  the  atmos- 
phere by  the  gases  of  combustion  transmitted  through  the  hot  iron. 
(See  Comptes  Rendus,  Jan.  20,  1868.  Report  of  commission  on  the 
observations  of  Dr.  Garret,  on  the  deleterious  effects  of  cast  iron 
stoves.) 

Porosity  of  Liquids.  If  measured  volumes  of  water  and  of  alco- 
hol are  mixed,  it  is  found  that  their  combined  volume  is  less  than 
the  sum  of  their  volumes  before  mixing.  The  greatest  contrac- 
tion takes  place  when  100  volumes  of  water  are  combined  with  116 
volumes  of  anhydrous  alcohol,  the  diminution  in  volume  being 
equal  to  3*7  per  cent. 

This  experiment,  due  to  Reaumur,  is  made  as  follows  :  to  a  bottle 
is  adapted  a  deeply  fitting  cork,  perforated  by  a  glass  tube.  The. 
8 


58  Lecture- Notes  on  Physics. 

requisite  proportion  of  water  is  placed  in  the  bottle,  and  on  this, 
colored  alcohol  is  gently  poured  until  it  completely  fills  the  bottle. 
The  stopper  is  now  forced  in,  and  the  alcohol  rises  in  the  glass  tube. 
The  height  of  the  liquid  is  marked,  and  the  bottle  being  so  inclined 
that  the  mixture  of  the  liquids  takes  place,  the  fluids  contract,  and 
the  alcohol  descends  in  the  tube. 

Contraction  also  takes  place  when  water  is  mixed  with  sul- 
phuric acid,  with  salt  and  with  other  substances.  No  experiment 
could  more  clearly  show  that  the  atoms  of  water  and  alcohol  do  not 
really  fill  the  space  which  the  liquids  appear  to  occupy. 

"Water  and  other  liquids  dissolve  gases;  for  example,  1  volume 
of  water  at  60°  Fah.  will  dissolve  720  volumes  of  ammonia, 
and  increase  in  its  volume  only  one-half,  and  in  its  weight  one- 
third. 

Porosity  of  gases.  Six  parts  by  weight  of  carbon  will  unite  with 
eight  parts  by  weight  of  oxygen,  and  yet  the  resulting  carbonic 
acid  gas  has  the  volume  of  the  oxygen  before  combination. 

Air  and  vapor  are  porous  to  each  other,  and  between  the  atoms 
of  air  there  may  exist  the  ultimate  parts  of  other  gases. 

It  is  to  be  remarked  that  this  property  of  porosity  is  not  opposed 
to  the  property  of  impenetrability,  for  it  is  the  atoms  which  are 
impenetrable,  while  it  is  the  inter  molecular  spaces  of  one  body 
which  are  penetrated  by  the  atoms  of  another. 

We  must  be  careful  not  to  confound  the  true  meaning  of  porosity, 
as  given  above,  with  what  we  designate  as  organic,  or  structural 
porosity.  Organic  pores  are  interstices  which  are  visible  with  the 
aid  of  the  microscope,  and  often  even  with  the  naked  eye,  while  the 
intermolecular  spaces  are  invisible  by  any  means. 

Examples  of  organic  porosity.  Mercury  forced  through  wood. 
Circulation  of  sap  in  plants. 

Examples  of  structural  porosity.  Porosity  of  stones,  caused  by 
the  minute  interstices  produced  by  the  crystalline  structure  of  the 
stone.  Paper  used  to  filter  liquids. 

8.  Mobility. 

Everywhere  in  the  universe,  we  see  masses  of  matter  changing 
their  positions  with  reference  to  each  other ;  while  others  appear  to 
be  at  rest. 

We  only  know  that  a  body  changes  its  position  by  referring  it 


Lecture- Notes  on  Physics.  59 

to  other  bodies,  supposed  to  be  at  rest ;  and  therefore  Descartes 
defined  motion  as  a  rectilinear  change  of  distance  between  two 
points.  From  this  definition,  if  only  one  point  existed  in  the  uni- 
verse, its  conditions,  whether  of  rest  or  of  motion,  could  not  be 
determined. 

If  the  points  to  which  we  refer  the  moving  body  be  really  at 
rest,  then  the  successive  varying  measures  give  us  the  absolute  motion, 
but  if  this  point  be  only  apparently  at  rest,  we  obtain  the  relative 
motion.  Give  illustrations. 

Rest  is  however  only  apparent ;  we  know  of  no  point  in  the  uni- 
verse which  is  absolutely  at  rest ;  therefore  all  the  motions  which 
we  have  determined  are  relative. 

All  bodies  on  the  surface  of  the  earth  have  the  relative  positions 
of  their  atoms  changed  by  every  change  of  temperature,  by  every 
vibration  which  they  transmit;  they  are  in  motion  with  the  earth 
on  its  axis  and  in  its  orbit,  while  the  earth  with  the  planets  and 
sun  are  being  translated  in  an  unknown  path  in  space.  The  stars, 
improperly  called  fixed,  have  also  motions  of  their  own,  which,  in 
some  cases,  are  evidently  of  great  magnitude. 

Kind  of  motions.  A  motion  may  be  one  of  translation,  of  rotation 
round  an  axis,  or  of  a  combination  of  these  motions ;  it  may  be  rec- 
tilinear or  curvilinear,  continued  or  reciprocating.  Give  illustrations. 

The  line  a  point  of  a  body  describes  in  space,  is  called  its  trajectory. 

The  velocity  of  motion  of  a  moving  body,  or  the  ratio  of  the  space 
described  to  the  time  of  describing  it,  may  be  uniform,  uniformly 
accelerated,  uniformly  retarded,  or  irregular. 

Uniform  motion  is  that  in  which  equal  spaces  are  passed  over  in 
equal,  small  portions  of  time. 

Formulae  of  uniform  motion. 

L;  s          s 

(1)     s  =  VT;  T  =  S- ;  V==- 

in  which  s  =  space  described ;  v  =  velocity  of  moving  body,  or  the 
space  described  in  unit  of  time ;  T= time  occupied  in  going  over  space  s. 

Uniformly  varied  motion  is  that  in  which  the  change  in  velocity 
at  the  end  of  a  certain  time,  is  proportional  to  this  time. 

Let  u  be  the  initial  velocity  of  the  body,  that  is  to  say,  the  velocity 
at  a  given  instant  from  which  we  commence  to  reckon  the  time ;  a 
the  acceleration,  or  the  constant  quantity  by  which  the  velocity  varies 


60  Lecture-  Notes  on  Physics. 

in  a  unit  of  time  ;  v  the  velocity  at  the  end  of  t  seconds  ;  we  have 
from  definition  of  uniformly  varied  motion. 

(2)     v  =  u  ±  a  t 

The  sign  -f-  when  the  motion  is  accelerated  ;  the  sign  —  when  it  is 
retarded.  In  the  last  case,  the  velocity  will  become  o  when  u=at, 
that  is  to  say,  at  the  end  of  a  number  of  seconds  represented  by  u  :  a. 

If,  in  formula  (2),  we  make  u  =  o,  then  the  moving  body  starts 
from  a  state  of  rest,  and  we  have  at  the  end  of  time  t 

(3)    v=at; 

that  is,  the  velocity  acquired  at  the  end  of  a  certain  time  is  proportional 
to  this  time. 

In  uniformly  accelerated  motion,  the  spaces  described  ~by  a  body  start- 
ing from  a  state  of  rest,  are  proportional  to  the  squares  of  the  times 
employed  to  describe  them;  or  — 

(4)  s  =  JcU2 

which  shows  that  the  space  gone  over  in  uniformly  accelerated 
motion,  by  a  body  which  starts  from  a  state  of  rest,  is  equal  to  the 
space  which  it  would  go  over  with  a  uniform  motion  with  the  velocity 
J  v  or  J  a  t. 

It  is  often  required  to  know  the  velocity  acquired  in  function  of 
the  space  gone  over.  We  obtain  this  by  eliminating  t  in  the  two 
equations  v  =  a  t,  s  =  J  at2,  which  gives  — 

(5)  v  = 


A  body  falling  vertically,  in  vacuo,  by  the  action  of  gravity,  from 
a  height  which  is  exceedingly  small  compared  with  the  radius  of  the 
earth,  may  be  regarded  as  having  a  uniformly  accelerated  motion. 
In  this  case  a  is  equal  -to  the  velocity  acquired  by  the  body  at  the 
end  of  the  first  second  of  its  fall.  This  velocity  which  is  the  measure 
of  the  intensity  of  gravity,  and  which  is  equal,  in  the  latitude  of  New 
York,  to  32  feet  2  inches,  is  always  indicated  by  the  letter  g.  The 
formula  (5)  then  becomes  — 


Problem.   "What  velocity  will  a  body  acquire  by  falling  772  feet, 
in  the  latitude  of  New  York?     Ans,  222  feet,  10-28  inches. 


Lecture- Notes  on  Physics.  61 

When  the  moving  body  possesses  an  initial  velocity  w,  at  the 
moment  from  which  we  count  the  time,  the  above  formulae  become — • 
v  =  u  ±  a  t\  s  —  u  t  +  J  a  t2,  v  =  yV  +~2a  s 

To  Galileo  is  due  the 'discovery  of  the  laws  of  uniformly  varied 
motion.  Give  his  classical  demonstration  of  these  laws. 

Co-existence  of  separate  motions.  When  a  body  in  motion  is  acted' 
on  by  a  force,  the  same  effect  in  motion  is  produced  as  if  that  force 
moved  the  body  from  a  state  of  rest.  Thus,  a  cannon  ball  pro- 
jected horizontally  from  an  elevation,  will  reach  the  ground  by 
the  action  of  gravity,  in  the  same  time  (assuming  that  no  air  resists 
its  motion). as  if  it  dropped  vertically  from  the  elevation  through 
the  same  height. 

"A  body  describes  the  diagonal  of  a  parallelogram  by  two  forces 
acting  conjointly,  in  the  same  time  in  which  it  would  describe  its 
sides  by  the  same  forces  acting  separately." — Newton. 

The  discovery  of  the  law  of  the  co-existence  of  motions,  is  also 
generally  ascribed  to  Galileo  (Dial.  4,  Prop.  2),  but  Aristotle  clearly 
announced  it  in  his  Mechanical  Problems,  c.  24. 

The  various  relations  of  space  and  time  can  readily  be  repre- 
sented geometrically,  by  making  lengths  of  ordinates  stand  for 
velocities  acquired  at  the  end  of  times  represented  by  lengths  of 
abscissas.  Thus,  uniformly  varied  motion  is  represented  by  a  right 
angled  triangle,  whose  base  equals  in  units  of  length  the  units  of 
time  during  which  the  motion  took  place,  its  altitude,  the  velocity 
acquired  at  the  end  of  this  time,  while  the  units  of  area  of  the 
triangle  represents  the  distance  gone  through  by  the  moving  body. 

The  consideration  of  motions,  irrespective  of  force  and  of  the 
properties  of  matter,  constitutes  the  part  of  science  called  Kinematics. 
(Gr.  Kerens,  motion.)  The  ablest  discussion  of  this  subject  is  found 
in  first  chapter  of  the  Natural  Philosophy,  by  Profs.  Thomson  and 
Tait,  Oxford,  1867. 

9.  Inertia — Force. 

The  property  of  matter  by  which  it  tends  to  retain  its  state, 
whether  of  rest  or  of  motion,  is  called  inertia. 

By  saying  that  a  body  has  inertia,  we  merely  understand  that  a 
body  cannot  of  itself  modify  its  condition,  whether  of  rest  or  of 
motion ;  and  that  whenever  a  body  begins  to  move,  or  to  change 
the  velocity  or  the  direction  of  its  motion,  these  changes  in  its  con- 
dition are  to  be  referred  to  some  extraneous  cause. 


62  Lecture-Notes  on  Physics. 

When  a  body  is  set  in  motion  and  abandoned  entirely  to  itself — 
when  it  is  conceived  as  being  alone  in  space — it  will  move  in  a 
straight  line,  which  is  the  direction  of  its  first  motion,  and  with  its 
first  velocity  forever.  This  truth,  called  the  law  of  inertia,  is  the 
result  of  an  extended  induction,  and  was  not  recognized  before  the 
time  of  Kepler.  Descartes  made  it  the '  foundation  of  his  princi- 
ples of  mechanics. 

Give  illustrations  of  above  principle,  from  observations  of  the 
motions  of  the  heavenly  bodies,  and  from  experiments  on  the 
motions  of  bodies  on  the  surface  of  the  earth.  The  rotation  of  the 
earth  on  its  axis.  A  pendulum  will  vibrate  two  days  in  a  vacuum, 
when  the  friction  of  the  point  of  suspension  is  reduced  to  its 
minimum. 

The  apparent  departures  from  the  law  of  inertia,  can  all  be  re- 
ferred to  the  action  of  forces  or  of  resistances  exterior  to  the  body, 
and  which  are  opposed  to  its  uniform  motion  in  a  straight  line. 

Force. 

All  the  phenomena,  or  changes  which  we  observe  in  the  condi- 
tion of  matter,  are  motions  or  the  results  of  motions  of  either  masses 
or  of  their  ultimate  parts  or  atoms;  and  that  which  produces  these 
changes  in  the  condition  of  matter  is  denominated  force. 

To  Dr.  Julius  Robert  Mayer,  of  Heilbronn,  Germany,  we  owe  the 
first  successful  attempt  to  give  as  clear  conceptions  in  reference  to 
force,  as  previously  existed  in  relation  to  matter.  In  184:2,  he  pub- 
lished in  Liebig'sAnnalen,  a  short  paper  of  eight  pages,  entitled  Bemer- 
kungen  uber  die  Krdfte  der  unbelebeten  Natur,  which  from  the  funda- 
mental importance  of  the  truths  which  it  unfolds,  and  from  the  re- 
sults which  have  been  deduced  from  them,  is  to  be  considered  as 
one  of  the  most  important  additions  to  knowledge  produced  in  this 
century. 

Mayer  reasons  thus:  " Forces  are  causes;  accordingly,  we  may 
in  relation  to  them,  make  full  application  of  the  principle,  causa, 
&quat  effectum.  If  the  cause  c  has  the  effect  e,  then  c  =  e;  if,  in  its 
turn,  e  is  the  cause  of  a  second  effect,  /,  we  have  e  — /,  and  so  on ; 
c  =  e  =f  .  .  .  .  =  c.  In  a  chain  of  causes  and  effects,  a  term  or  a 
part  of  a  term  can  never,  as  plainly  appears  from  the  nature  of  an 
equation,  become  equal  to  nothing.  This  first  property  of  all  causes 
we  call  their  indestructibility. 

"  If  the  given  cause,  c,  has  produced  an  effect,  e,  equal  to  itself,  it 


Lecture- Notes  on  Physics.  63 

has  in  that  very  act  ceased  to  be;  c  has  become  e;  if,  after  the  pro- 
duction of  e,  c  still  remained  in  whole  or  in  part,  there  must  be 
still  further  effects  corresponding  to  this  remaining  cause ;  the  total 
effect  of  c  would  thus  be  >e,  which  would  be  contrary  to  the  sup- 
position c  =  e.  Accordingly,  since  c  becomes  e,  and  e  becomes  /, 
&c.,  we  must  regard  these  various  magnitudes  as  different  forms 
under  which  one  and  the  same  object  makes  its  appearance.  This 
capability  of  assuming  various  forms,  is  the  second  essential  pro- 
perty of  all  causes.  Taking  both  properties  together,  we  rnay 
say  causes  are  (quantitatively)  indestructible  and  (qualitatively)  con- 
vertible objects." 

In  another  important  paper,  " Bemerkungen  ilber  das  mechanische 
Aequivalent  der  Warme,  1851,"  Mayer  says:  "Force  is  something 
which  is  expended  in  producing  motion  ;  and  this  something  which 
is  expended  is  to  be  looked  upon  as  a  cause  equivalent  to  the  effect, 
namely,  to  the  motion  produced." 

Now,  in  these  motions  or  effects,  there  are  evidently  two  things  to 
be  considered,  (1)  the  mass  of  matter  moved,  and  (2)  the  space 
through  which  it  is  moved;  and  we  have  therefore  force  =  mass X 
space  gone  through  ;  but  as  we  can  measure  and  compare  forces  only 
by  measuring  and  comparing  their  effects,  and  as  bodies  in  free 
motion  will  move  in  the  same  right  line,  and  with  uniform  velocity 
forever,  we  must  place  the  moving  bodies  in  such  circumstances 
that  their  motions  are  destroyed,  and  we  have  remaining  in  their 
stead,  equivalent  effects,  which  we  can  measure;  then  the  compari- 
son of  these  measures  will  give  us  the  relative  intensities  of  the 
forces. 

These  effects,  either  directly  or  indirectly  obtained  from  the  mov- 
ing body,  are  as  various  as  there  are  kind  of  forces  and  resistances 
existing;  thus,  we  may  oppose  to  a  body,  moving  vertically  up- 
ward, the  resistance  of  gravity  (which  we  may  regard  as  constant, 
if  the  upward  flight  of  the  body  is  a  very  minute  fraction  of  the 
radius  of  the  earth) ;  or,  we  may  oppose  the  constant  resistance, 
which  a  body  of  homogeneous  structure  offers  when  it  is  penetrated 
by  another,  as,  for  example,  when  a  cannon  ball  penetrates  earth, 
clay,  or  pine  wood ;  or,  again,  we  may  have,  for  the  effect  of  the 
destroyed  motion,  heat,  which  makes  its  appearance  whenever  a 
moving  body  is  brought  to  rest  either  by  friction,  percussion,  com- 
pression, &c.,  with  some  other  body,  or,  as  in  Foucault's  experi- 
ment, where  a  copper  disk  being  forced  to  revolve  between  the 


64  Lecture-Notes  on  Physics. 

poles  of  a  powerful  electro-magnet,  the  motion  of  the  wheel  being 
opposed  by  the  reaction  existing  between  the  electric  currents  flow- 
ing in  it  and  in  the  magnet,  the  motion  (force)  lost  by  the  wheel 
appears  as  heat  (force)  in  its  substance. 

The  heat  which  is  produced  by  any  of  these  means  can  readily  be 
caused  to  evolve,  as  it  disappears,  dynamic  electricity,  light,  and 
chemical  action.  Thus,  in  an  experiment  which  the  author  devised 
for  his  classes,  about  four  years  ago,  the  heat  developed  by  the  "fall- 
ing force  "  of  a  weight  striking  the  terminals  of  a  compound  thermal- 
battery,  formed  of  pieces  of  iron  wire  and  German  silver  wire  twisted 
together  at  alternate  ends,  caused  a  current  of  electricity  through 
the  wires,  which  being  conducted  through  a  helix  magnetized  a 
a  needle  (which  then  attracted  fine  iron  particles),  caused  light  to 
appear  in  a  portion  of  the  circuit  formed  of  Wollaston's  fine  wire, 
decomposed  iodide  of  potassium,  and  finally  moved  the  needles  of  a 
galvanometer. 

Let  us  now  try  to  arrive  at  comparable  measures  of  force  by  first 
opposing  to  the  moving  body  the  constant  resistance  of  gravity, 
and  see  if  the  measures  thus  given  for  different  velocities  compare 
with  measures  given  by  other  resistances  overcome,  and  for  differ- 
ent quantities  of  heat,  electricity,  &c.,  developed  by  the  disappear- 
ance of  different  velocities  in  the  moving  mass. 

A  body,  shot  vertically  upward,  with  a  velocity  vt  comes  to  rest, 
by  the  opposing  resistance  of  gravity,  after  having  reached  a  certain 
height,  which  we  will  call  /?.  Giving  the  body  twice  the  initial 
velocity,  2v,  it  reaches  4h  before  it  begins  to  return  to  the  earth ; 
with  the  initial  velocity  3?;,  it  reaches  the  height  9A;  while  4v  gives 
16A,  and  so  on;  in  other  words,  the  heights  reached  are  as  the  squares 
of  the  initial  velocities. 

Wherefore,  as  force  =  mass  X  space  gone  through,  it  follows  that 
the  measure  of  force  is  mass  X  v2;  v  being  the  velocity  of  the  mass; 
or,  force  =  mass  X  distance  gone  through  in  overcQming  the  constant 
resistance  =mass  X  v2. 

If  this  measure  be  true,  the  same  ratio  of  the  square  of  the 
velocity,  will  exist  when  other  resistances  are  opposed  to  the  mov- 
ing body.  Take  the  resistance  offered  by  an  earth  or  clay  bank  to 
the  penetration  of  cannon  balls  having  different  velocities.  It  is 
found  by  artillerists,  that  a  ball  striking  with  the  velocity  2t>,  will 
penetrate  four  times  as  deep  as  the  same  ball  with  velocity  v;  while 
a  velocity  of  3v  will  give  a  penetration  of  nine  times  the  depth  and 


Lecture-Notes  on  Physics.  65 

so  on ;  the  penetration  of  the  same  ball  being  as  the  squares  of  its 
velocities.  (See  Dr.  Wollaston's  Bakerian  Lecture  on  the  Force  of 
Percussion,  Phil.  Trans.  1806;  and  Benton's  Ordnance  and  Gun- 
nery, K  Y.,  1862,  p.  476,  et  seq.) 

Having  found  this  measure  of  force  true  for  these  two  cases, 
where  motion  disappears,  let  us  determine,  as  we  only  can,  experi- 
mentally and  inductively,  whether  the  relative  quantities  of  heat 
evolved  as  different  velocities  of  the  moving  mass  disappear,  also 
preserve  the  ratio  of  the  squares  of  those  velocities. 

In  the  year  1850,  'there  appeared  in  the  Phil.  Trans.  E.  S.  Lond., 
a  paper  "On  the  Mechanical  Equivalent  of  Heat"  by  Dr.  James 
Preseott  Joule,  of  Manchester,  England.  This  memoir  contains 
one  of  the  most  important  physical  constants  ever  determined. 

In  this  investigation  was  first  obtained,  by  direct  experiment,  the 
exact  quantity  of  heat  developed  by  the  falling  of  a  given  weight 
through  a  known  height.  His  experiments  on  this  subject  began 
in  1843,  and  were  continued  during  six  years,  until  1849.  Dr. 
Joule,  during  this  long  experimental  experience,  gradually  per- 
fected his  apparatus,  and  learned  to  eliminate  various  sources  of 
error,  until  finally  his  measures  arrived  at  by  different  processes 
gave,  within  small  limits,  almost  identical  values  for  "  the  mechani- 
cal equivalent  of  heat."* 

His  apparatus  consisted  of  an  upright  copper  cylinder,  which 
contained  either  water,  oil,  or  mercury ;  in  the  lid  of  this  vessel 
were  two  apertures,  one  for  a  vertical  axis,  to  revolve  in  without 
touching  the  lid,  the  other  for  the  introduction  of  a  thermometer. 
The  vertical  axis,  which  was  perfectly  fitted  into  the  bottom  of  the 
vessel,  carried  eight  revolving  arms  or  paddles,  which,  as  they  went 
round,  passed  between  openings  in  four  stationary  vanes,  so  that  the 
water  could  not  acquire  a  motion  of  rotation  and  move  with  the 
arms;  and  resistance  was  thus  made  to  their  motion.  Two  weights 
were  attached  to  fine  flexible  cords,  which  passed  over  pulleys,  and 
were  wound  round  a  roller  on  the  vertical  axis,  armed  with  the 
paddles.  These  weights  in  falling  caused  the  paddles  to  revolve, 
and  by  the  resistance  which  it  opposed  to  their  motion,  the  liquid 
was  heated  by  the  equivalent  in  motion  expended.  The  height  of 

*  It  is  to  be  remarked  that  Mayer,  in  1842,  used  the  expression  "  mechanical 
equivalent  of  heat,"  and  from  the  difference  in  the  specific  heats  of  the  same  weight 
of  air  under  constant  pressure,  and  under  constant  volume,  theoretically  deduced  a 
value  for  this  equivalent,  which,  corrected  with  the  exact  data  of  the  above  quan- 
tities as  furnished  by  Kegnault,  gives  a  result  nearly  identical  with  Joule's. 


66 


Lecture-Notes  on  Physics. 


the  fall  was  about  sixty-three  feet  (to  which  we  may  say  that  practi- 
cally the  radius  of  the  earth  was  infinite),  and  was  measured  by 
vertical  scales,  along  which  the  weights  descended. 

The  mode  of  experimenting  was  as  follows  :  the  temperature  of 
the  liquid  having  been  ascertained  by  a  thermometer,  which  was 
capable  of  indicating  a  variation  of  temperature  as  small  as  2Jotn 
of  a  degree  Fah.,  and  the  weights  wound  up  by  detaching  the  roller 
from  the  vertical  paddle-axis,  the  precise  height  of  the  weights  were 
ascertained  after  keying  the  roller;  the  weights  then  descended  until 
they  reached  the  floor.  The  roller  was  again  detached  from  the 
paddle-axis,  the  weights  wound  up,  and  the  agitation  of  the  liquid 
renewed.  This  was  repeated  twenty  times,  and  then  the  tempera- 
ture of  the  liquid  was  again  observed.  The  mean  temperature  of 
the  room  was  derived  from  observations  made  at  the  beginning, 
middle,  and  end  of  each  experiment. 

Corrections  were  now  made  for  the  effects  of  radiation  and  con- 
duction ;  and,  in  the  experiments  with  water,  for  the  quantities  of 
heat  absorbed  by  the  copper  vessel  and  by  the  paddle  wheel.  In 
the  experiments  on  the  heat  produced  by  the  agitation  of  mercury, 
and  in  the  heat  given  out  by  the  rubbing  of  cast  iron  plates,  the 
heat  capacity  of  the  entire  apparatus  was  ascertained  by  observ- 
ing the  heating  which  it  produced  on  a  known  weight  of  water  in 
which  it  was  immersed.  In  all  the  experiments,  corrections  were 
also  made  for  the  velocity  with  which  the  weights  came  to  the 
ground,  and  for  the  rigidity  of  the  strings.  The  force  expended  in 
friction  of  the  apparatus  was  diminished  as  far  as  possible  by  the 
use  of  friction  wheels,  and  its  amount  was  determined  by  connect- 
ing the  pulleys  without  connection  with,  the  paddle-axis,  and  ascer- 
taining the  weight  necessary  to  give  them  a  uniform  motion. 

The  following  table  gives  the  results  of  Joule;  the  second  col- 
umn, as  they  were  obtained  in  air,  the  third,  the  same  corrected,  as 
though  the  weights  had  descended  in  vacuo. 


MATERIALS  EMPLOYED. 

MEAN  EQUIV. 
IN  AIR. 

MEAN  EQUIV. 
IN  VAC. 

NO.   OF    EXP'S 
FROM  WHICH 
DERIVED. 

Water  

773-640 

772-602 

40 

775-032 

774-083 

50 

Cast  Iron  

775-938 

774-987 

20 

Lecture-Notes  on  Physics.  67 

In  the  experiments  of  producing  heat  by  the  rotation  of  one 
cast  iron  plate  on  another,  the  friction  produced  considerable  vibra- 
tion in  the  frame  work  of  the  apparatus,  and  a  loud  sound ;  allow- 
ance was  therefore  made  for  the  quantity  of  force  expended  in  pro- 
ducing these  effects. 

The  number  772'692,  obtained  as  the  mean  of  forty  experiments 
on  the  friction  of  water,  Joule  considered  the  most  trustworthy ; 
but  this  he  reduced  to  772,  because,  even  in  the  friction  of  liquids, 
he  found  it  impossible  entirely  to  avoid  vibration  and  sound.  The 
deductions  of  Joule  from  these  experiments  are  : 

1.  That  the  quantity  of  heat  produced  by  the  friction  of  bodies, 
whether   solid  or    liquid,    is    always  proportional   to    the  force   ex- 
pended. 

2.  That  the  quantity  of  heat  capable  of  increasing  the  temperature 
of  one  pound  of  water  (weighed  in  vacuo,  and  between  55°  and  60°) 
by  1°  Fah.,  requires  for  its  evolution  the  expenditure  of  a  mechanical 
force  represented  bj  the  fall  of  1  pound  through  772  feet,  or  772  foot- 


This  is  the  "  Mechanical  Equivalent  of  Heat"  or  the  unit  of  heat, 
generally  called  in  honor  of  the  illustrious  physicist,  "Joule's 
Unit." 

In  French  measures,  the  above  heat-unit  is  thus  stated :  The  heat 
capable  of  increasing  the  temperature  of  1  kilogramme  of  water  1°  C., 
is  equivalent  to  a  force  represented  by  the  fall  of  423'55  kilogrammes, 
through  the  space  of  1  metre.  The  descent  or  ascent  of  1  pound 
through  1  foot  is  called  afoot-pound,  while  the  descent  or  ascent  of 
1  kilogramme  through  1  metre  is  denominated  a  kilo gramme-metre. 
By  the  adoption  of  these  terms,  the  expressions  of  the  above  truths 
can  be  mere  concisely  enunciated;  thus,  using  French  measures, 
we  say,  423  kilogramme-metres  is  equivalent  to  1  kilogramme-degree 
centigrade. 

Thus,  Joule  showed  that  the  heat  developed  was  in  proportion  to 
the  mass  X  the  distance  fallen  through ;  or,  what  is  the  same,  equiva- 
lent to  the  mass  X  square  of  the  velocity. 

In  a  remarkably  interesting  paper,  "  On  the  Production  of  Thermo^ 
Electric  Currents  by  Percussion,"  Prof.  O.  K.  Eood,  of  Columbia 
College,  N.  Y.,  shows  directly  by  careful  and  skilful  experiments,, 
that  the  heat  and  its  equivalent  in  dynamic  electricity  (which  latter 
gave  the  measure  of  the  heat),  produced  by  a  weight  falling  from 
different  heights  on  a  compound  plate  of  German  silver  and  iron,j 


68 


Lecture- Notes  on  Physics. 


was  in  proportion  to  the  height  of  the  fall  of  the  weight,  or,  what 
is  the  same,  to  the  square  of  the  velocity  of  impact. 

For  the  details  of  these  experiments,  and  the  precautions  taken 
to  avoid  the  action  of  extraneous  causes  mingling  themselves  with 
the  main  effect,  the  reader  is  referred  to  Prof  Eood's  paper  in  "The 
American  Journal  of  Science,  July,  1866." 

We  here  only  give  some  of  the  results,  which  speak  for  them- 
selves. 


TABLE  2.  —  WITH  Two  SKINS  ABOVE  AND  BELOW  JUNCTION  OF  METALS. 

1  in. 

2  ins. 

2°-7 

3  ins. 
3°-55 

4  ins. 
5-°2 

Deflection  of  Galvanometer-needles 
—  Average  of  eight  Observations 

TABLE  3.  —  WITH  FOUR  LAYERS  OF  PLAIN  SILK  ABOVE  AND  BELOW  THE 
JUNCTURE. 

Distances  Fallen  

lin. 
l°-5 

2  ins. 
3°-0 

3  ins. 
40.5 

4  ins. 
6°-0 

Average  Deviation  of  eight  exp'ts. 

TABLE  4.  —  WITH  FOUR  LAYERS  OF  WAX-COATED  WOVEN  SILK. 

Distances  Fallen  

1  in. 
1°-07 

2  ins. 
2°-l 

3  ins. 
3°  -28 

4  ins. 
4°  -3 

5  ins. 
6°-0 

Keduced  Average  Deviation  of  Five 

For  a  proper  interpretation  of  the  above  results,  it  is  to  be  re- 
marked that  the  deflection  of  the  galvanometer  needles  up  to  6° 
was  in  proportion  to  the  intensity  of  the  current,  which  is  itself, 
within  these  limits,  proportional  to  the  heat  at  the  juncture  which 
developed  it. 

It  can  be  further  shown  that  the  measure  of  force,  or  energy  (a 
term  now  more  generally  used,  and  which  we  owe  to  Dr.  Thomas 
Young),  is  proportional  to  the  square  of  the  velocity  of  a  moving 


Lecture-Notes  on  Physics 


mass,  by  obtaining  the  equivalents  of  effects  other  than  those  of 
resistance  offered  by  gravity,  by  penetrable  bodies,  and  of  the  heat 
developed  by  friction  and  by  impact.  Joule,  in  1843,  showed  that 
the  same  relation  existed  between  the  heat  evolved  by  the  electric 
current  of  an  electro-magnetic  engine,  and  the  mechanical  energy 
expended  in  producing  it,  and  in  1844  he  showed  that  the  heat  ab- 
sorbed and  evolved  by  the  rarefaction  and  condensation  of  air,  is 
proportional  to  the  amount  of  mechanical  energy  evolved  and  ab- 
sorbed in  these  operations. 

In  the  following  table,  taken  from  Yerdet's  "  Expose  de  la  Theorie 
MecJianique  de  la  Chaleur^  Paris,  1863,  are  given  the  most  reliable 
determinations  of  the  mechanical  equivalent  of  heat.  The  numbers 
represent  the  number  of  kilogramme-metres,  which  is  equivalent 
to  one  kilogramme-degree  centigrade  of  water. 


>-'  ^  ' 

*3 

M    U    2 

gS 

^   H   W 

^ 

EH 

£;  w  H 

W    pq 

i 

•<   pj   3 

5 

pa 

W   O    H 

W    ^ 

jj 

NATURE  OP  THE  PHENOMENON 

see 

K  S 

u 

WHENCE    THE    DETERMINA- 

a ^ 

^  H 

& 

TION  IS  DRAWN. 

§  K  « 

02    „ 

H 

|  H  H 

P5    £*j 

ri 

SgS^ 

r  . 

0 

o  2  o  o 

^ 

hr    ^     S 

1/2  W  "^ 

5 

rs^ 

p2 

1 

General  Properties  of  Air  

J  Mayer 

Y.  Eegnault,  1 
Moll&Van  [ 

426 

1  Clausius 

Beek            J 

Joule 

424 

Joule 

Favre 

413 

"Work  done  by  the  Steam  Engine 

Clausius 

Him 

413 

Heat  evolved  by  Induced  Currents 

Joule 

Joule 

452 

Heat  evolved    by  an   Electro-  ~| 

Magnetic  Engine  at  Rest  and  > 

Favre 

Favre 

443 

in  Motion  J 

Total  Heat  evolved  in  the  cir-  ) 
cuit  of  a  Daniell's  Battery...  / 

Bosscha 

fw.  Weber  | 
j      Joule        J 

420 

Heat  evolved  in  a  metallic  wire,  ") 

through  which  an  electric        I 

Clausius 

Quintus  Icilius 

400 

current  is  passing  J 

From  the  above  discussion,  we  conclude  that  the  true  measure  of 
force  is  mass  X  v2. 


70  Lecture-Notes  on  Physics. 

The  Theory  of  Energy  considers  its  Conservation,  its  Transfor- 
mation and  its  Dissipation. 

The  Conservation  of  Energy.— The  total  amount  of  energy  in  the 
universe,  or  in  any  limited  system  which  does  not  receive  energy 
from  without,  or  part  with  it  to  external  matter,  is  invariable. 
Energy  is,  in  other  words,  as  indestructible  as  matter,  and  is  neither 
created  nor  destroyed,  but  merely  changes  its  form. 

The  Transformation  of  Energy. — By  an  extended  induction,  we 
find  that  any  one  form  of  energy  may  be  transformed,  wholly  or 
partially,  to  an  equivalent  amount  in  another  form.  These  trans- 
formations are,  however,  subject  to  limitations  contained  in  the 
principle  of 

The  Dissipation  of  Energy.  —  "No  known  natural  process  is 
exactly  reversible,  and  whenever  an  attempt  is  made  to  transform 
or  retransform  energy  by  an  imperfect  process,  part  of  the  energy 
is  necessarily  transformed  into  heat  and  dissipated,  so  as  to  be 
incapable  of  further  useful  transformation.  It  therefore  follows 
that  as  energy  is  constantly  in  a  state  of  transformation,  there  is  a 
constant  degradation  of  energy  to  the  final  unavailable  form  of 
uniformly  diffused  heat ;  and  this  will  go  on  as  long  as  transforma- 
tions occur,  until  the  whole  energy  of  the  universe  has  taken  this 
final  form."  See  N.  Brit.  Rev.,  May,  1864. 

"  There  is  consequently,"  says  Prof.  Thomson,  "  so  far  as  we  under- 
stand the  present  condition  of  the  universe,  a  tendency  towards  a 
state  in  which  all  physical  energy  will  be  in  the  state  of  heat,  and 
that  heat  so  diffused  that  all  matter  will  be  at  the  same  temperature ; 
so  that  there  will  be  an  end  of  all  physical  phenomena." 

Yast  as  this  speculation  may  seem,  it  appears  to  be  soundly 
based  on  experimental  data,  and  to  represent  truly  the  present  con- 
dition of  the  universe,  so  far  as  we  know  it.  See  Prof.  Thomson 
U0n  a  Universal  Tendency  in  Nature  to  the  Dissipation  of  Mechanical 
Energy."— Proc.  K.  S.  Edinb.  and  Phil.  Mag.,  1852. 

The  energy  of  a  moving  body  is  the  work  which  it  is  capable  of 
performing  against  a  resistance  before  being  brought  to  rest,  and  is 
equal  to  the  force  which  must  act  on  the  body  to  move  it  from  a 
state  of  rest  to  the  given  velocity.  This  force  is  measured  by  the 
product  of  the  mass  of  the  body  into  the  height  from  which  it 
must  fall  in  order  to  acquire  the  given  velocity ;  which  is  expressed 
thus: 

M  V2 


Lecture-Notes  on  Physics.  71 

g  representing  as  usual,  the  velocity  acquired  by  a  body  at  the  end 
of  the  first  second  of  its  fall. 

Energy  may  be  of  two  kinds  (1),  Kinetic  energy,  or  energy  of 
motion,  and  (2)  Potential  energy,  or  energy  of  position  or  of  condi- 
tion. Thus,  the  energy  of  a  ball  shot  vertically  upward,  is  entirely 
kinetic  at  the  moment  of  its  discharge,  while  its  energy  is  all 
potential,  or  one  of  position,  when  it  has  reached  the  summit  of 
its  flight  and  begins  to  descend.  It  is  evident  that  the  ball  in 
descending,  will  gradually  lose  potential  energy  and  gain  kinetic 
energy  (the  sum  of  the  two  energies  always  remaining  constant), 
and  when  it  has  reached  the  level  from  which  it  was  discharged 
upward,  its  energy  is  again  all  kinetic,  and  equal  to  what  it  was 
when  it  began  its  upward  flight.  An  oscillating  pendulum  is  an 
instance  where  the  energy  is  alternately  kinetic  and  potential. 

All  the  various  forms  of  energy  may  be  brought  under  two 
classes. 

I.    Visible  Energy,  or  energy  of  visible  motions' and  positions. 
II.  Molecular  Energy. 

Under  class  I.  we  have 

A.  Visible  kinetic  energy. 

B.  Potential  energy  of  visible  arrangement,  as  for  examples, 

a  head  of  water ;  a  coiled  spring ;  a  raised  weight. 
Under  class  II. 

C.  The  energy  of  electricity  in  motion. 

D.  The  energy  of  radiant  heat  and  light. 
E-  The  kinetic  energy  of  absorbed  heat. 
F.  Molecular  potential  energy. 

Gr.  Potential  energy  caused  by  electrical  separation. 
H.  Potential  energy  caused  by  chemical  separation. 
Now  with  regard  to  these  various  forms  of  energy,  the  principle 
of  the  conservation  of  energy  asserts  that  for  a  body  left  to  itself, 
or  for  the  entire  material  universe,  we  must  have 

A-+-  B  +  C  -f  D  +  &c., =a  constant  quantity. 

On  the  other  hand,  the  various  terms  of  the  left  hand  member  of 
this  equation  must  be  considered  as  variable  quantities,  subject, 
however,  to  the  above  limitation,  but  capable  of  being  trans- 
formed into  one  another  according  to  certain  laws. 

Laws  of  the  transmutations  of  energy. — The  following  are  among 


72  Lecture-Notes  on  Physics. 

the  most  important  cases  of  transmutation  of  these  energies  into 
one  another. 

A  into  B,  when  a  weight  is  projected  upwards ;  into  C,  when  a 
conductor  revolves  between  the  poles  of  a  magnet.  A  is  not  trans- 
muted directly  into  D ;  it  is  into  E,  F  and  Gr.  A  is  not  directly  con- 
verted into  H. 

B  can  be  converted  into  A,  and  through  it  into  other  forms  of 
energy. 

C  can  be  transmuted  into  A,  into  E,  into  F,  and  into  H. 

D  can  be  transmuted  into  E,  into  F,  and  into  H. 

E  and  F  is  converted  into  A  and  into  B,  in  the  action  of  any 
heat-engine ;  into  C,  into  D  ;  into  Gr  when  tourmalines  are  heated ; 
and  into  H. 

Gr  can  be  converted  into  A  and  into  C. 

H  can  be  transmuted  into  C ;  into  E,  into  F,  and  into  G.  (See 
Elementary  Treatise  on  Heat,  by  Balfour  Stewart,  Oxford,  1866.) 

Sources  of  Energy. — The  energy  available  for  the  production  of 
mechanical  work  is  almost  entirely  potential,  and  consists  of 

Potential  forms  of  Energy. 

1.  Energy  of  Fuel. 

2.  Energy  of  Food. 

3.  Energy  of  Ordinary  Water-Power. 

4.  Energy  of  Tidal  Water-Power. 

5.  Energy  of   Chemical  separation  implied  in  native  sulphur, 

native  metals,  free  oxygen,  &c. 

. 

Of  Kinetic  forms  of  Energy. 

6.  Energy  of  Winds  and  Ocean  Currents. 

7.  Energy  of  Direct  Eays  of  the  Sun. 

8.  Energy  of  Volcanoes,  Hot  Springs  and  Internal  Heat  of  the 
Earth. 

"The  immediate  sources  of  these  supplies  of  energy  are  four : — 

I.  Primordial  Potential  Energy  of  Chemical  Affinity,  which 

probably  still  exists  in  native  metals,  native  sulphur,  &c., 

but  whose  amount,  at  all  events  near  the  surface  of  the 

earth,  is  now  very  small. 

II.  Solar  Kadiation.       III.  The  Earth's  rotation  about  its  axis. 
IV.  The  Internal  Heat  of  the  Earth. 


Lecture -Notes  on  Physics.  73 

Thus,  as  regards  (1)  our  supplies  of  fuel  for  heat-engines  are,  as 
was  long  ago  remarked  by  Herschel  and  Stephenson,  mainly  due  to 
solar  radiation.  Our  coal  is  merely  the  result  of  transformation 
in  vegetables,  of  solar  energy  into  potential  energy  of  chemical 
affinity.  So,  on  a  small  scale,  are  diamond,  amber  and  other  com- 
bustible products  of  primeval  vegetation.  As  Prof.  Thomson 
remarks,  wood  fires  give  us  heat  and  light  which  have  been  got 
from  the  sun  a  few  years  ago.  Our  coal  fires  and  gas  lamps  bring 
out,  for  our  present  comfort,  heat  and  light  of  a  primeval  sun, 
which  have  lain  dormant  as  a  potential  energy,  beneath  seas  and 
mountains  for  countless  ages. 

Though  (II)  thus  accounts  for  the  greater  part  of  our  store  of 
energy,  (I)  must  also  be  admitted,  though  to  a  very  subordinate  place. 

As  to  (2),  the  food  of  all  animals  is  vegetable  or  animal,  and 
therefore  ultimately  vegetable.  This  energy,  then,  depends  almost 
entirely  on  (II).  This,  also,  was  stated  long  ago  by  Herschel. 

Ordinary  water-power  (3)  is  the  result  of  evaporation,  the  diffu- 
sion and  convection  of  vapor,  and  its  subsequent  condensation  at  a 
higher  level.  It  also  is  mainly  due  to  (II). 

Tidal  water-power  (4),  although  not  yet  much  used,  is  capable,  if 
properly  applied,  of  giving  valuable  supplies  of  energy.  As  the 
water  is  lifted  by  the  attraction  of  the  sun  and  moon,  it  may  be 
secured  by  proper  contrivances  at  its  higher  level,  and  then  becomes 
an  available  supply  of  energy  when  the  tide  has  fallen  again.  Any 
such  supply  is,  however,  abstracted  from  the  energy  of  the  earth's 
rotation  (III).  This  was  recognized  by  Kant;  Mayer  also  and  J. 
Thomson  showed  that  the  ebb  and  flow  of  the  tides  being  due  to 
the  earth's  revolving  on  her  axis  under  the  moon's  attraction,  the 
energy  of  the  tides  is  really  taken  from  the  energy  of  the  earth's 
revolution ;  part  of  which  is  thus  ultimately  dissipated  in  the  heat 
of  friction  caused  by  the  tides.  The  general  tendency  of  tides  on 
the  surface  of  a  planet  is  to  retard  its  rotation  till  it  turns  always 
the  same  face  to  the  tide-producing  body ;  and  it  is  probable  that 
the  remarkable  fact  that  satellites  generally  turn  the  same  face  to 
their  primary,  is  to  be  accounted  for  by  tides  produced  by  the  pri- 
mary in  the  satellite  while  it  was  yet  in  a  molten  state. 

"Winds  and  ocean  currents  (6),  both  employed  in  navigation,  and 
the  former  in  driving  machinery,  are,  like  (3),  direct  transformations 
of  solar  radiation  (II). 

10 


74  Lecture- Notes  on  Physics. 

As  to  (8),  which  is  due  to  (IV),  no  application  to  useful  mechanical 
purposes  has  yet  been  attempted. 

We  must  next  very  briefly  consider  the  origin  of  these  causes, 
with  the  exception  of  (I),  which  is  of  course  primary.  Laplace, 
Mayer  and  Helmholtz  come  to  our  assistance,  and  suggest  as  the 
initial  form  of  the  energy  of  the  universe,  the  potential  energy  of 
gravitation  of  matter  irregularly  diffused  through  infinite  space. 
By  simple  calculations  it  is  easy  to  see  that,  if  the  matter  in  the 
solar  system  had  been  originally  spread  through  a  space  enclosing 
the  orbit  of  Neptune,  the  falling  together  of  its  parts  into  separate 
agglomerations,  such  as  the  sun  and  planets,  would  far  more  than 
account  for  all  the  energy  they  now  possess  in  the  forms  of  heat 
and  orbital  and  axial  revolutions. 

The  sun  still  retains  so  much  potential  energy  among  its  parts, 
that  the  mere  contraction  by  cooling  must  be  sufficient  (on  ac- 
count of  the  diminution  of  potential  energy)  to  maintain  the 
present  rate  of  radiation  for  ages  to  come.  Moreover,  the  capacity 
of  the  sun's  mass  for  heat,  on  account  especially,  of  the  enormous 
pressure  to  which  it  is  exposed,  is  so  great  that  (at  the  least  and 
most  favorable  assumption)  from  7,000  to  8,000  years  must  elapse, 
at  the  present  rate  of  expenditure,  before  the  temperature  of  the 
whole  is  lowered  one  degree  centigrade,  although  the  amount  of 
solar  heat  received  by  the  earth  in  one  year  is  so  enormous  that  it 
would  liquify  a  layer  of  ice  100  feet  thick,  covering  the  whole 
surface  of  the  earth,  and  if  we  bear  in  mind  that  the  solar  heat 
which  reaches  the  earth  in  any  time  is  only  ssouuouotj^j  of  the  heat 
which  leaves  the  sun,  we  may  obtain,  some  idea  of  the  immense 
heating  power  of  the  radiation  from  our  luminary.* 

It  thus  appears  that  if  we  except  tidal-power,  the  sun's  rays  are 
the  ultimate  source  of  the  available  forms  of  energy  with  which  we 
are  surrounded.f 

We  see,  from  the  above  exposition,  that  "  Philosophers  have 
extended  their  ideas  of  quantity  from  matter  to  energy,  and  thus 
has  arisen  the  new  science  of  Energetics,  or  the  quantitative  study 
of  the  transformations  of  energy  (as  chemistry  is  the  quantitative 
study  of  the  transformations  of  matter),  comprehending  and  uniting 
all  the  different  branches  of  physical  science." 

*  If  the  entire  solar  radiation  were  employed  in  dissolving  a  layer  of  ice,  enclos- 
ing the  sun,  it  would  dissolve  a  stratum  10J  miles  thick  in  a  day. 

f  The  sketch  we  here  give  of  the  Sources  of  Energy,  is  taken  almost  entirely 
from  an  article  on  "  Energy,'  in  the  N.  Brit.  Rev.,  May,  1864. 


Lecture- Notes  on  Physics.  75 

Efficiency  of  Heat- Engines. — After  Joule  had  determined  the 
mechanical  equivalent  of  heat,  engineers  had  the  means  of  testing 
the  actual  efficiency  of  heat-engines. 

If  the  number  of  thermal  units  produced  by  the  combustion  of 
one  pound  of  a  given  kind  of  fuel,  be  multiplied  by  Joule's  unit, 
772  foot-poundsT  the  result  is  the  total  heat  of  combustion  of  the 
given  fuel  expressed  in  foot-pounds.  This  quantity  ranges  between 
5,000,000  and  12,000,000  foot-pounds.  But  in  the  best  existing 
steam-engines,  it  is  found  that  on  an  average  only  about  J-  of  the 
mechanical  value  of  the  heat  produced  by  the  fuel  burning  in  the 
furnace,  is  obtained  as  useful  mechanical  effect,  the  remaining  f 
being  wholly  lost. 

To  understand  the  cause  of  this  great  loss,  it  is  to  be  remembered 
that  in  every  heat-engine  the  heat  of  the  expansible  fluid — which 
is  the  medium  by  which  the  heat  of  the  fuel  is  transformed  into  the 
motion  of  the  engine — disappears  as  heat  by  the  exact  equivalent, 
expressed  in  Joule's  units,  of  the  motion  produced.  Therefore  the 
greater  the  fall  in  heat  in  the  vapor,  which,  in  expanding,  cools  and 
gives  up  its  heat  as  motion,  so  will  be  the  efficacy  of  the  engine. 
Just  as  in  a  head  of  water,  where  the  greater  the  difference  between 
the  higher  and  lower  level,  the  greater  the  power  obtained.  But  in 
the  steam-engine  we  are  obliged  to  obtain  our  power  from  the  fall 
in  temperature  of  the  steam,  which  takes  place  in  its  expansion,  so 
that  in  a  steam-engine  the  power  obtained  is  measured  by  the  differ- 
ence of  temperature  between  the  boiler  and  condenser,  and  not  by 
the  difference  between  the  temperature  of  the  furnace  and  con- 
denser. Now,  in  the  furnace  the  temperature  is  about  3,000  degrees 
above  that  of  the  atmosphere,  while  the  temperature  of  the  boiler 
is  only  about  200  degrees  in  excess  of  that  of  the  condenser ;  there- 
fore it  is  evident  that  the  larger  fall  in  temperature  taking  place 
between  the  furnace  and  the  steam,  the  heat  is  lost  or  at  least  not 
utilized. 

In  a  perfect  engine  the  steam  would  enter  the  cylinder  at  the  tem- 
perature of  the  furnace,  and  expand  down  until  it  had  given  up  all 
its  heat  as  motion  to  the  piston,  and  would  then  enter  the  condenser 
at  the  temperature  of  the  atmosphere ;  indeed,  such  an  engine  would 
require  no  condenser,  for  the  steam  would  condense  itself  as  the 
heat  disappeared  in  its  transmutation  into  motion. 

We  will  conclude  this  sketch  on  the  subject  of  Energy,  with  a 
concise  statement  in  reference  to  the  efficiency  of  heat-engines 


76  Lecture-Notes  on  Physics. 

taken  from  Prof.  Eankine ;  referring  the  reader  who  desires  further 
information  on  this  important  and  interesting  subject  to  the  works 
given  below. 

"  The  total  heat  produced  in  the  furnace  is  expended,  in  any 
given  engine,  in  producing  the  following  effects,  whose  sum  is  equal 
to  the  heat  so  expended : — 

1.  The  waste  heat  of  the  furnace,  being  from  0*1  to  0*6  of  the 
total  heat,  according  to  the  construction  of  the  furnace  and  the  skill 
with  which  the  combustion  is  regulated. 

2.  The  necessarily -rejected  heat  of  the  engine,  being  the  excess  of 
the  whole  heat  communicated  to  the  working  fluid  by  each  pound 
of  fuel  burned,  above  the7  portion  of  that  heat  which  permanently 
disappears,  being  replaced  by  mechanical  energy. 

3.  The  heat  wasted  ly  the  engine,  whether  by  conduction  or  by 
non-fulfilment  of  the  conditions  of  maximum  efficiency. 

4.  The  useless  work  of  the  engine,  employed  in  overcoming  friction 
and  other  prejudicial  resistances. 

5.  The  useful  work.    The  efficiency  of  a  heat-engine  is  improved 
by  diminishing  as  far  as  possible,  the  first  four  of  those  effects,  so 
as  to  increase  the  fifth. 

"  It  appears,  then,  that  the  efficiency  of  a  heat-engine  is  the  pro- 
duct of  three  factors,  viz  :  I.  The  efficiency  of  the  furnace,  being  the 
ratio  which  the  heat  transferred  to  the  working  fluid  bears  to  the 
total  heat  of  combustion ;  II.  The  efficiency  of  the  fluid,  being  the 
fraction  of  the  heat  received  by  it,  which  is  transformed  into 
mechanical  energy ;  and  III.,  The  efficiency  of  the  mechanism,  being 
the  fraction  of  that  energy  which  is  available  for  driving  machines." 

From  the  above  discussion,  we  see  immediately  that  by  super- 
heating the  steam  before  it  reaches  the  cylinder,  we  obtain  a  greater 
range  of  temperature  for  the  steam  to  fall  through  in  expanding, 
and  thus  render  efficacious  yet  more  of  the  heat  of  the  furnace. 

List  of  Works  on  the  Conservation  of  Force  and  Thermodynamics. 

The  Correlation  and  Conservation  of  Forces ;  a  collection  of  the 
papers  of  Mayer,  Helmholtz,  Faraday,  Grove,  Liebig  and  Carpenter. 
Edited  by  Dr.  Edward  L.  Youmans,  N.  Y.,  1865. 

Joule. — On  the  Calorific  Effects  of  Magneto-Electricity,  and  on 
the  Mechanical  Yalue  of  Heat.  Phil.  Mag.,  Vol.  XXIII.,  1843. 

On  the  Changes  of  Temperature  produced  by  the  Earefaction  and 
Condensation  of  Air.  Phil.  Mag.,  May,  1845. 


Lecture-Notes  on  Physics.  77 

On  the  Mechanical  Equivalent  of  Heat.     Phil.  Trans.,  1850. 

On  some  Thermodynamic  Properties  of  Solids.  Phil.  Trans., 
1859. 

On  the  Thermal  Effects  of  Compressing  Fluids.  Phil.  Trans., 
1859. 

Clausius. — The  Mechanical  Theory  of  Heat,  with  its  applications 
to  the  Steam  Engine  and  to  the  Physical  Properties  of  Bodies. 
Edited  by  T.  A.  Hirst,  with  an  Introduction  by  Prof.  Tyndall. 
London,  1867. 

Thomson  (William). — An  Account  of  Carnot's  Theory  of  the 
Motive  Power  of  Heat.  Trans.  K.  S.,  Edinb.,  1849. 

On  the  Dynamical  Theory  of  Heat.     Trans.  E.  S.,  Edinb.,  1852. 

Thomson  and  Joule. — On  the  Thermal  Effects  of  Fluids  in  Motion. 
Phil.  Trans.,  1853. 

On  the  Changes  of  Temperature  experienced  by  Bodies  moving 
through  Air.  Phil.  Trans.,  1860. 

Rankine. — The  Steam  Engine  and  other  Prime  Movers.  London, 
1859. 

Verdel. — Expose  de  la  Thdorie  Mechanique  de  la  Chaleur.  Paris, 
1863. 

Him. — Exposition  Analytique  et  Experimentale  de  la  Theorie 
Mechanique  de  la  Chaleur.  Paris,  1865. 

Saint — Robert. — Principes  de  Thermodynamique.     Turin,  1865. 

Bacon. — Novum  Organum,  De  Forma  Calidi,  book  2,  aph.  20. 

"  Now  from  this  our  first  vintage  it  follows,  that  the  form  or  true  definition  of 
heat  (considered  relatively  to  the  universe  and  not  to  the  sense)  is  briefly  thus  : — 
Heat  is  a  motion,  expansive,  restrained,  and  acting  in  its  strife  upon  the  smaller 
particles  of  bodies.  But  the  expansion  is  thus  modified  :  while  it  expands  all  ways, 
it  has  at  the  same  time  an  inclination  upwards.  And  the  struggle  in  the  particles 
is  modified  also;  it  is  not  sluggish,  but  hurried  and  with  violence." 

Locke  — "Heat  is  a  very  brisk  agitation  of  the  insensible  parts  of  the  object, 
which  produce  in  us  that  sensation  from  whence  we  denominate  the  object  hot;  so 
what  in  our  sensation  is  heat,  in  the  object  is  nothing  but  motion." 

Rumford. — Trans.  R.  S.  Lond.,  1798.  Kumford  placed  a  cannon  in  a  water- 
tight box,  so  that  it  could  rotate  against  a  blunt  borer  firmly  pressed  against  the 
bottom  of  its  chamber.  The  box  was  filled  with  water  of  a  temperature  of  60°  F., 
and  the  cannon  set  in  rotation  by  the  power  of  horses. 

"  The  result  of  this  beautiful  experiment  was  very  striking,  and  the  pleasure  it 
afforded  me  amply  repaid  me  for  all  the  trouble  I  had  had  in  contriving  and 
arranging  the  complicated  machinery  used  in  making  it.  The  cylinder  had  been 
in  motion  but  a  short  time,  when  I  perceived,  by  putting  my  hand  into  the  water, 
and  touching  the  outside  of  the  cylinder,  that  heat  was  generated. 

"At  the  end  of  an  hour  the  fluid,  which  weighed  18-77  Ibs.,  or  2^  gallons,  had 
its  temperature  raised  47  degrees,  being  now  107  degrees. 


78  Lecture-Notes  on  Physics. 

"  In  thirty  minutes  more,  or  one  hour  and  thirty  minutes  after  the  machinery 
had  been  set  in  motion,  the  heat  of  the  water  was  142  degrees. 

"At  the  end  of  two  hours  from  the  beginning,  the  temperature  was  178  degrees. 

"At  two  hours  and  twenty  minutes  it  was  200  degrees,  and  at  two  hours  and 
thirty  minutes  it  actually  boiled. 

"  From  the  results  of  my  computations,  it  appears  that  the  quantity  of  heat 
produced  equably,  or  in  a  continuous  stream,  if  I  may  use  the  expression,  by  the 
friction  of  the  blunt  steel  borer  against  the  bottom  of  the  hollow  metallic  cylinder, 
was  greater  than  that  produced  in  the  combustion  of  nine  wax  candles,  each  f  inch 
in  diameter,  all  burning  together  with  clear  bright  flames. 

"One  horse  would  have  been  equal  to  the  work  performed,  though  two  were 
actually  employed.  Heat  may  thus  be  produced  merely  by  the  strength  of  a  horse, 
and,  in  case  of  necessity,  this  heat  might  be  used  in  cooking  victuals.  But  no 
circumstances  could  be  imagined  in  which  this  method  of  procuring  heat  would  be 
advantageous  ;  for  more  heat  might  be  obtained  by  using  the  fodder,  necessary  for 
the  support  of  a  horse,  as  fuel. 

*  *    *    *    *     "  It  is  hardly  necessary  to  add,  that  anything  which  any  insulated 
body  or  system  of  bodies  can  continue  to  furnish  without  limitation,  cannot  possibly 
be  a  material  substance;  and  it  appears  to  me  to  be  extremely  difficult,  if  not 
quite  impossible,  to  form  any  distinct  idea  of  anything  capable  of  being  excited 
and  communicated  in  those  experiments,  except  motion" 

2)avy. — First  scientific  memoir,  entitled  "On  Heat,  Light,  and 
the  Combinations  of  Light."  "Works  Vol.  II. 

"  Experiment. — I  procured  two  parallelopipedons  of  ice,  of  the  temperature  of 
29°,  six  inches  long,  two  wide,  and  two-thirds  of  an  inch  thick  ;  they  were  fastened 
by  wires,  to  two  bars  of  iron.  By  a  peculiar  mechanism,  their  surfaces  were 
placed  in  contact,  and  kept  in  a  continued  and  most  violent  friction  for  some 
minutes.  They  were  almost  entirely  converted  into  water,  which  water  was 
collected,  and  its  temperature  ascertained  to  be  35°,  after  remaining  in  an  atmos- 
phere of  a  lower  temperature  for  some  minutes.  The  fusion  took  place  only  at  the 
plane  of  contact  of  the  two  pieces  of  ice,  and  no  bodies  were  in  friction  but  ice. 

"  From  this  experiment  it  is  evident  that  ice  by  friction  is  converted  into  water 
and  according  to  the  supposition,  its  capacity  is  diminished  ;  but  it  is  a  well-known 
fact  that  the  capacity  of  water  for  heat  is  much  greater  than  that  of  ice ;  and  ice 
must  have  an  absolute  quantity  of  heat  added  to  it  before  it  can  be  converted  into 
water.  Friction  consequently  does  not  diminish  the  capacity  of  bodies  for  heat. 

*  ****** 

"  Now  a  motion  or  vibration  of  the  corpuscules  of  bodies  must  be  necessarily 
generated  by  friction  and  percussion.  Therefore  we  may  reasonably  conclude  that 
this  motion  or  vibration  is  heat,  or  the  repulsive  power. 

Davy  in  his  Chemical  Philosophy,  p.  95,  says : 

<«  *  *  *  *  The  immediate  cause  of  the  phenomena  heat,  then,  is  motion,  and 
the  laws  of  its  communication  are  precisely  the  same  as  the  laws  of  the  communi- 
cation of  motion." 

A  process  similar  to  the  above  classical  experiment  of  Davy,  has  from  time 
immemorial  been  used  by  savage  nations  in  obtaining  fire  by  means  of  friction: 
not  by  rubbing  together  sticks,  as  usually  stated,  for  no  one  could  thus  produce 
ignition,  but  by  whirling  rapidly,  a  pointed  rod  against  a  wooden  block,  by  means 
of  an  arrangement  similar  to  the  watchmaker's  bow  and  drill.  It  is  worthy  of 
remark  that  this  apparatus  has  everywhere  the  same  form,  whether  used  by  the 


Lecture-Notes  on  Physics.  79 

islanders  of  the  Pacific  or  by  the  aborigines  of  our  own  country.     One  of  these 
instruments  can  be  seen  in  the  State  Cabinet  of  Natural  History  of  New  York. 

Henry  (Dr.  Joseph). —  Meteorology — Patent  Office  Keport  for 
1857. 

On  Acoustics  applied  to  Public  Buildings — Smithsonian  Eeport, 
1856,  p.  228. 

"The  tuning-fork  was  next  placed  upon  a  cube  of  India  rubber,  and  this  upon 
the  marble  slab.  The  sound  emitted  by  this  arrangement  was  scarcely  greater 
than  in  the  case  of  the  tuning-fork  suspended  from  the  cambric  thread,  and  from 
the  analogy  of  the  previous  experiments,  we  might  at  first  thought  suppose  the 
time  of  duration  would  be  great;  but  this  was  not  the  case.  The  vibrations  con- 
tinued only  about  forty  seconds.  The  question  may  here  be  asked,  what  became 
of  the  impulses  lost  by  the  tuning-fork?  They  were  neither  transmitted  through 
the  India  rubber  nor  given  off  to  the  air  in  form  of  sound,  but  were  probably 
expended  in  producing  a  change  in  the  matter  of  the  India  rubber,  or  were  con- 
verted into  heat,  or  both.  Though  the  inquiry  did  not  fall  strictly  within  the 
line  of  this  series  of  investigations,  yet  it  was  of  so  interesting  a  character  in  a 
physical  point  of  view  to  determine  whether  heat  was  actually  produced,  that  the 
following  experiment  was  made. 

*  *  *  «  And  the  point  of  a  compound  wire,  formed  of  copper  and  iron,  was 
thrust  into  the  substance  of  the  rubber,  while  the  other  ends  of  the  wire  were 
connected  with  a  delicate  galvanometer.  The  needle  was  suffered  to  come  to  rest, 
the  tuning-fork  was  then  vibrated,  and  its  impulses  transmitted  to  the  rubber.  A 
very  perceptible  increase  of  temperature  was  the  result.  The  needle  moved  through 
an  arc  of  from  one  to  two  and  a  half  degrees.  The  experiment  was  varied,  and 
many  times  repeated;  the  motions  of  the  needle  were  always  in  the  same  direc- 
tion, namely,  in  that  which  was  produced  when  the  point  of  the  compound  wire 
was  heated  by  momentary  contact  with  the  fingers.  The  amount  of  heat  generated 
in  this  way  is,  however,  small,  and  indeed,  in  all  cases  in  which  it  is  generated  by 
mechanical  means,  the  amount  evolved  appears  very  small  in  comparison  with  the 
labor  expended  in  producing  it." 

Leibnitz. — "  The  force  of  a  moving  body  is  proportional  to  the  square  of  its 
velocity,  or  to  the  height  to  which  it  would  rise  against  gravity." 

Wollaston. — Bakerian  Lecture  on  the  Force  of  Percussion.  Phil. 
Trans.,  Yol.  XCYL,  1806. 

"  In  short,  whether  we  are  considering  the  sources  of  extended  exertion  or  of  accu- 
mulated energy,  whether  we  compare  the  accumulated  forces  themselves  by  their 
gradual  or  by  their  sudden  effects,  the  idea  of  mechanic  force  in  practice  is  always 
the  same,  and  is  proportional  to  the  space  through  which  any  moving  force  is 
exerted  or  overcome,  or  to  the  square  of  the  velocity  of  a  body  in  which  such  force 
is  accumulated." 

Tynddll. — Heat  considered  as  a'  Mode  of  Motion.  New  York, 
1863. 

I  need  hardly  refer  the  reader  to  this  charming  exposition  of 
Prof.  Tyndall ;  who,  by  his  enthusiasm  and  vividness  of  illustration, 
has  rendered  his  subject  so  popular  that  his  work  is,  in  this  country, 
in  the  library  of  nearly  every  man  of  culture. 


80  Lecture-Notes  on  Physics. 

10.  Attraction,  11.  Repulsion. 

Attraction  exists  between  the  minute  parts  or  atoms  of  bodies, 
and  they  would  continually  approach,  until  contact  supervened,  if 
they  were  not  kept  apart  by  an  equal  and  opposing  repulsion. 
These  molecular  attractions  and  repulsions  are  often  designated  as 
the  molecular  forces. 

The  phenomena  of  traction,  of  compression,  of  porosity,  and  of 
the  transmission  of  vibrations  through  all  kind  of  matter,  prove 
indirectly  that  bodies  are  formed  of  minute  parts  which  do  not 
touch,  but  are  kept  at  certain  distances  depending  on  the  intensity 
of  the  attractions  and  repulsions  subsisting  between  them;  while 
there  are  many  direct  proofs  of  the  above  statement  which  will  be 
given  further  on. 

All  masses  of  matter  mutually  attract  each  other  with  an  inten- 
sity directly  proportional  to  their  masses,  and  inversely  as  the 
squares  of  their  distances.  This  tendency  is  called  gravitation,  and 
is  common  to  all  matter.  This  is  shown  in  the  celebrated  experi- 
ment devised  by  the  Eev.  John  Michell,  and  generally  known  as 
the  Cavendish  experiment  for  determining  the  density  of  the  earth. 
Describe  this  apparatus. 

Electric  and  magnetic  attractions  and  repulsions  also  follow  the 
law  of  the  inverse  squares  of  the  distances. 

The  different  manifestations  of  attraction  and  repulsion  may  be 
thus  arranged : 

GRAVITATION,  which  act  at  scnsible  distances .  L  e, 

the  **W*  of  an  inch, 


MAGNETISM, 

COHESION, 

ADHESION,  I  Which  act  at  insensible  distances ;  i.  e. 

CAPILLARITY,  within  the  ^oo111  of  an  incl1- 

CHEMICAL  AFFINITY,  j 

Molecular  Attraction. 

Cohesion  designates  the  attraction  existing  between  the  minute 
parts  of  the  same  body ;  adhesion  the  attraction  between  the  parts 
of  dissimilar  bodies.  Sometimes  designated  respectively  as  homo- 
geneous and  heterogeneous  attraction. 

Experiments  on  Cohesion  of  solids. — Two  leaden  planes  pressed 
together,  cohere  with  a  force  of  forty  pounds  to  the  square  inch. 
Two  plates  of  glass  cohere  even  in  vacuo,  which  shows  that  the 


Lecture- Notes  on  Physics.  81 

phenomenon  is  not  due  to  the  atmospheric  pressure.  The  intensity 
of  the  cohesion  in  this  experiment  is  proportional  to  the  surface, 
and  increases  with  the  time  of  contact.  In  plate  glass  manufacto- 
ries, mirror  glasses  sometimes  cohere  with  such  force,  from  having 
been  placed  on  each  other  without  intervening  paper,  that  it  is 
impossible  to  separate  them.  It  is  to  be  remarked  that  in  the  above 
instances  only  a  comparatively  few  points  of  the  cohering  surfaces 
are  in  contact. 

"  There  is  a  precious  experiment  by  Mr.  Huyghens  in  No.  86  of 
the  Philosophical  Transactions.  A  piece  of  mirror  glass  being  laid 
on  the  table,  and  another,  to  which  a  handle  was  cemented  on  one 
surface,  being  gently  pressed  on  it,  with  a  little  of  a  sliding  motion, 
the  two  adhered,  and  the  one  lifted  the  o^her.  Lest  this  should  have 
been  produced  by  the  pressure  of  the  atmosphere,  Mr.  Huyghens 
repeated  the  experiment  in  an  exhausted  receiver,  with  the  same 
success. 

u  He  found  that  one  plate  carried  the  other,  although  they  were 
not  in  mathematical  contact,  but  had  a  very  sensible  distance  between 
them.  He  found  this  by  wrapping  round  one  of  the  plates  a  single 
fibre  of  silk  drawn  off  from  the  cocoon.  The  adhesion  was  vastly 
weaker  than  before,  but  still  sufficient  for  carrying  the  lower  plate. 

"  Here,  then,  is  a  most  evident  and  and  incontrovertible  example 
of  a  mutual  attraction  acting  at  a  distance.  Mr.  Huyghens  found 
that  if,  in  wrapping  the  fibre  round  the  glass,  he  made  it  cross  a  fibre 
already  wrapped  round  it,  there  was  no  sensible  attraction.  In  this 
case,  the  glasses  were  separated  by  a  distance  equal  to  twice  the 
diameter  of  a  fibre  of  silk. 

"  I  said  that  this  experiment  showed  that  it  was  not  the  attrac- 
tion of  gravitation  that  produced  the  cohesion.  I  have  repeated 
the  experiment  with  the  most  scrupulous  care,  measuring  the  dis- 
ance  of  the  glasses  (the  diameter  of  a  silk  fibre),  and  the  weight 
supported.  I  find  this,  in  all  cases,  to  be  nearly  14  J  times  the  action 
of  gravity.  The  calculation  is  obvious  and  easy.  I  tried  it  in  dis- 
tances considerably  different,  according  to  the  diameter  oT  the  fibre. 
I  must  inform  the  person  who  would  derive  his  information  from 
his  own  experiments,  that  there  are  many  circumstances  to  be  at- 
tended to  which  are  not  obvious,  and  which  materially  affect  the 
result.  The  silk  fibres  are  not  round,  but  very  flat,  one  diameter 
being  almost  double  of  the  other.  The  2400th  part  of  an  inch  may 
be  considered  as  the  average  smaller  diameter  of  a  fibre.  A  mag- 
11 


82  Lecture-Notes  on  Physics. 

nif ying  glass  must  be  used,  and  great  patience  in  wrapping  the  fibre 
round  the  glass  so  that  it  may  not  be  twisted.  A  flaxen  fibre  is 
much  preferable,  when  gotten  single,  and  fine  enough,  for  it  is  a 
perfect  cylinder.  I  must  also  -inform  him,  that  no  regularity  will 
be  had  in  experiments  with  bits  of  ordinary  mirror;  these  are 
neither  flat  enough,  nor  well  enough  polished.  We  must  employ 
the  square  pieces  which  are  made  and  finished  by  a  very  few  Lon- 
don artists  for  the  specula  of  the  best  Hadley's  [Godfrey's]  quadrants. 
These  must  be  most  carefully  cleaned  of  all  dust  or  damp.  Yet  this 
must  not  be  done  by  wiping  them  with  a  clean  cloth ;  this  infal- 
libly deranges  everything  by  rendering  the  plate  electric.  I  suc- 
ceeded best  by  keeping  them  in  a  glass  jar,  in  which  a  piece  of 
moist  cloth  was  lying,  but  jaot  touching  the  glasses.  "When  wanted, 
the  glasses  are  taken  out  with  a  pair  of  tongs  and  held  a  little  while 
before  the  fire,  which  dissipates  the  damp  which  had  adhered  to 
them,  and  which  prevented  all  electricity.  With  these  precautions, 
and  a  careful  measurement  of  the  diameter  of  the  silk  fibre,  the  ex- 
periments will  rarely  differ  among  themselves  one  part  in  ten. 

« *  *  *  *  *  *  If  the  plates  have  been  hard 
pressed,  with  a  sliding  or  grinding  motion,  the  adhesion  is  then 
either  very  strong,  or  nothing  at  all ;  when  they  do  adhere,  it  seems 
to  be  another  stage  or  alternation  of  the  force,  as  will  be  explained 
by  and  by.  But  they  rarely  adhere,  owing  to  fragments  torn  off 
by  the  grinding.  The  glasses  will  be  scratched  by  it. 

"  I  thought  this  capital  experiment  worthy  of  a  very  minute  de- 
scription, it  being  that  which  gives  us  the  means  of  mathematical 
and  dynamical  treatment  in  the  greatest  perfection."  "A  System  of 
Mechanical  Philosophy,  by  John  Eobison,  L.  L.  D.  Edited  by  Sir 
David  Brewster,  Edinburg:  printed  for  John  Murray,  London,  1822." 
Yol.  I.  page  240,  et  seq. 

Cohesion  in  solids  is  measured  by  the  force  in  pounds  avoirdu- 
pois required  to  tear  apart,  by  a  direct  pull,  a  rod  of  one  square 
inch  area  of  section.  This  measure  is  called  the  tenacity  of  a  body. 

TABLE  OF  TENACITIES  FROM  "KANKINE'S  APPLIED  MECHANICS." 

Steel -. 116.COO 

Iron,  wire 95,000 

"      wire  ropes 90,000 

u      wrought  bars  and  bolts 65,000 

«      cast 10,500 

Copper,  wire 60,000 


Lecture-Notes  on  Physics.  83 

Copper,  cast 19,fOO 

Brass,  wire 49.0')0 

"  cast 18,0>0 

Gun-metal  (copper  8,  tin  1) 36,000 

Zinc 7,50J 

Tin,  cast , 4,600 

Lead,  sheet 3,300 

Teak 18,00) 

Ash 17,000 

Mahogany 16,20) 

Locust 16,(H  0 

Oak,  European 14,900 

"  American  red  10,250 

Fir — red  pine '.? 13.*  00 

"      spruce 12,400 

"  larch 9,500 

Chestnut 12,0.0 

Beech 11,5(0 

Maple 10,600 

Hempen  cables 5,600 

Slate 11,2-10 

Glass 9,400 

Brick 290 

Mortar 50 

Cohesion  of  liquids  is  shown  in  the  force  required  to  separate  a 
disk  of  wood  from  a  liquid  which  wets  it ;  this  force  varies  with 
the  liquid,  requiring  52  grains  per  square  inch  to  separate  the  disk 
from  water;  31  grains  for  oil  of  turpentine;  28  grains  for  alcohol. 
These  experiments,  however,  as  will  be  seen  below,  give  only  the 
relative  cohesion. 

Prof.  Joseph  Henry,  in  a  valuable  contribution  to  molecular 
physics,  published  in  the  Proceedings  of  the  American  Philoso- 
phical Society  for  April,  1844,  showed  that  the  molecular  at- 
traction of  water  for  water,  instead  of  being  only  about  fifty-two 
grains  to  the  square  inch,  is  really  several  hundred  pounds,  and  is 
probably  equal  to  that  of  the  attraction  of  ice  for  ice.  The  follow- 
ing are  extracts  from  Dr.  Henry's  paper : 

"  The  passage  of  a  body  from  a  solid  to  a  liquid  state  is  generally 
attributed  to  the  neutralization  of  the  attraction  of  cohesion  by  the 
repulsion  of  the  increased  quantity  of  heat;  the  liquid  being  sup- 
posed to  retain  a  small  portion  of  its  original  attraction,  which  is 
shown  by  the  force  necessary  to  separate  a  surface  of  water  from 
water,  in  the  well  known  experiment  of  a  plate  suspended  from  a 
scale  beam  over  a  vessel  of  the  liquid.  It  is,  however,  more  in 
accordance  with  all  the  phenomena  of  cohesion  to  suppose,  instead 


84  Lecture-Notes  on  Physics. 

of  the  attraction  of  the  liquid  being  neutralized  by  the  heat,  that 
the  effect  of  this  agent  is  merely  to  neutralize  the  polarity  of  the 
molecules  so  as  to  give  them  perfect  freedom  of  motion  around 
every  imaginable  axis.  The  small  amount  of  cohesion  (52  grains 
to  the  square  inch),  exhibited  in  the  foregoing  experiment,  is  due, 
according  to  the  theory  of  capillarity  of  Young  and  Poisson,  to  the 
tension  of  the  exterior  film  of  the  surface  of  water  drawn  up  by 
the  elevation  of  the  plate.  This  film  gives  way  first,  and  the  strain 
is  thrown  on  an  inner  film,  which,  in  turn,  is  ruptured ;  and  so  on 
until  the  plate  is  entirely  separated ;  the  whole  effect  being  similar 
to  tearing  the  water  apart  atom  by  atom. 

"  Keflecting  on  the  subject,  the  author  has  thought  that  a  more 
correct  idea  of  the  magnitude  of  the  molecular  attraction  might  be 
obtained  by  studying  the  tenacity  of  a  more  viscid  liquid  than  water. 
For  this  purpose,  he  had  recourse  to  soap  water,  and  attempted  to 
measure  the  tenacity  of  this  liquid  by  means  of  weighing  the 
quantity  of  water  which  adhered  to  a  bubble  of  this  substance  just 
before  it  burst,  and  by  determining  the  thickness  of  the  film  from 
an  observation  of  the  color  it  exhibited  in  comparison  with  New- 
ton's scale  of  thin  plates.  Although  experiments  of  this  kind 
could  only  furnish  approximate  results,  yet  they  showed  that  the 
molecular  attraction  of  water  for  water,  instead  of  being  only  about 
52  grains  to  the  square  inch,  is  really  several  hundred  pounds,  and 
is  probably  equal  to  that  of  the  attraction  of  ice  for  ice.  The  effect 
of  dissolving  the  soap  in  the  water,  is  not,  as  might  at  first  appear, 
to  increase  the  molecular  attraction,  but  to  diminish  the  mobility 
of  the  molecules,  and  thus  render  the  liquid  more  viscid. 

"  According  to  the  theory  of  Young  and  Poisson,  many  of  the 
phenomena  of  liquid  cohesion,  and  all  those  of  capillarity,  are  due 
to  a  contractile  force  existing  at  the  free  surface  of  the  liquid,  and 
which  tends  in  all  cases  to  urge  the  liquid  in  the  direction  of  the 
radius  of  curvature  towards  the  centre,  with  a  force  inversely  as 
this  radius.  [The  explanation  of  the  existence  of  this  contractile 
force  will  be  given  in  the  next  section  of  the  Notes,  which  con- 
siders Capillarity.] 

"  According  to  this  theory,  the  spherical  form  of  a  dew-drop  is 
not  the  effect  of  the  attraction  of  each  molecule  of  the  water  on 
every  other,  as  in  the  action  of  gravitation  in  producing  the  globu- 
lar form  of  the  planets  (since  the  attraction  of  cohesion  only  extends 
to  an  appreciable  distance),  but  it  is  due  to  the  contractile  force 


Lecture-Notes  on  Physics.  85 

which  tends  constantly  to  enclose  the  given  quantity  of  water  within 
the  smallest  surface,  namely,  that  of  a  sphere.  The  author  finds  a 
contractile  force  similar  to  that  assumed  by  this  theory,  in  the  sur- 
face of  the  soap-bubble ;  indeed,  the  bubble  may  be  considered  a 
drop  of  water  with  the  internal  liquid  removed,  and  its  place  sup- 
plied by  air.  The  spherical  form  in  the  two  cases  is  produced  by 
the  operations  of  the  same  cause.  The  contractile  force  in  the  sur- 
face of  the  bubble  is  easily  shown  by  blowing  a  large  bubble  on 
the  end  of  a  wide  tube,  say  an  inch  in  diameter ;  as  soon  as  the 
rnouth  is  removed,  the  bubble  will  be  seen  to  diminish  rapidly,  and 
at  the  same  time  quite  a  forcible  current  of  air  will  be  blown 
through  the  tube  against  the  face.  This  effect  is  not  due  to  the 
ascent  of  the  heated  air  from  the  lungs,  with  which  the  bubble  was 
inflated,  for  the  same  effect  is  produced  by  inflating  with  cold  air, 
and  also  when  the  bubble  is  held  perpendicularly  above  the  face,  so 
that  the  current  is  downwards. 

"  Many  experiments  were  made  to  determine  the  amount  of  this 
force,  by  blowing  a  bubble  on  the  larger  end  of  a  glass  tube  in  the 
form  of  a  letter  U,  and  partially  filled  with  water ;  the  contractile 
force  of  the  bubble,  transmitted  through  the  enclosed  air,  forced 
down  the  water  in  the  larger  leg  of  the  tube,  and  caused  it  to  rise 
in  the  smaller.  The  difference  of  level  observed  by  means  of  a 
microscope,  gave  the  force  in  grains  per  square  inch,  derived  from 
the  known  pressure  of  a  given  height  of  water.  The  thickness  of 
the  film  of  soap-water  which  formed  the  envelope  of  the  bubble, 
was  estimated  as  before,  by  the  color  exhibited  just  before  burst- 
ing. The  results  of  these  experiments  agree  with  those  of  weigh- 
ing the  bubble,  in  giving  a  great  intensity  to  the  molecular  attrac- 
tion of  tne  liquid ;  equal  at  least  to  several  hundred  pounds  to  the 
square  inch.  Several  other  methods  were  employed  to  measure  the 
tenacity  of  the  film,  the  general  results  of  which  were  the  same ; 
the  numerical  details  of  these  are  reserved,  however,  until  the  ex- 
periments can  be  repeated  with  a  more  delicate  balance. 

"  The  comparative  cohesion  of  pure  water  and  soap- water  was 
determined  by  the  weight  necessary  to  detach  the  same  plate  from 
each ;  and  in  all  cases  the  pure  water  was  found  to  exhibit  nearly 
double  the  tenacity  of  the  soap-water.  The  want  of  permanency 
in  the  bubble  of  pure  water  is  therefore  not  due  to  feeble  attrac- 
tion, but  to  the  perfect  mobility  of  the  molecules,  which  causes  the 
equilibrium,  as  in  the  case  of  the  arch,  without  friction  of  parts,  to 
be  destroyed  by  the  slightest  extraneous  force." 


86  Lecture-Notes  on  Physics. 

The  above  investigation  of  Dr.  Henry  will  be  referred  to  again 
under  the  head  of  Capillarity. 

Gases. — Between  the  molecules  of  the  same  gas  repulsion  exists, 
but  a  slight  attraction  appears  to  prevail  between  the  molecules  of 
different  gases. 

Adhesion. — Of  solids  to  solids. — Placing  of  metals.  Gold  leaf  stamped 
on  metals.  "  It  is  known  that  if  two  pieces  of  metal  are  scraped 
very  clean,  a  severe  blow  will  make  them  to  cohere  so  as  to  be  in- 
separable. It  is  thus  that  flowers  of  gold  and  silver  are  fixed  on 
steel  and  other  metals.  The  steel  is  first  scraped  clean,  and  a  thin 
bit  of  gold  or  silver  is  laid  on  it,  and  then  the  die  is  applied  by  a 
strong  blow  with  a  hammer.  It  is  remarkable  that  they  will  not 
adhere  with  such  firmness,  if  they  adhere  at  all,  when  the  surfaces 
have  been  polished  in  the  usual  way,  with  fine  powders,  &c.  This 
is  always  done  with  the  help  of  greasy  matters.  Some  of  this  pro- 
bably remains,  and  prevents  that  specific  action  that  is  necessary. 
I  am  disposed  to  think  that  the  scraping  of  the  surfaces'  also  ope- 
rates in  another  way,  viz :  by  filling  the  surface  with  scratches, 
that  is,  ridges  and  furrows.  These  allow  the  air  to  escape  as  the 
pieces  come  together  by  the  blow.  If  the  mere  blow  were  sufficient, 
a  coin  would  adhere  fast  to  the  die.  But,  in  coining,  the  flat  face 
of  the  die  first  closes  with  the  piece  of  metal,  and  effectually  con- 
fines the  air  which  fills  the  hollow  that  is  to  form  the  relief  of  the 
coin.  This  air  must  be  compressed  to  a  prodigious  degree,  and  in 
this  state,  it  is  still  between  the  die  and  the  coin.  "We  may  say  that 
the  impression  on  the  coin  is  really  formed  by  this  included  air ; 
for  the  metal  in  this  part  of  the  coin  is  never  in  contact  with  the 
die.  I  know  of  two  cases,  which  greatly  confirm  this  conjecture. 
The  dies  chanced  to  crack  in  the  highest  part  of  the  relief,  and  after 
this  were  thrown  aside  (although  in  one,  for  a  common  die,  the  crack 
was  quite  insignificant),  because  the  coin  could  seldom  be  parted 
from  them." — [JRobison.~] 

"When  bladder  is  dried  on  glass,  the  adhesion  is  so  great  that  it 
cannot  be  torn  off  without  bringing  with  it  some  of  the  glass. 

Of  solids  to  liquids. — A  rapidly  issuing  jet  of  water  is  deflected 
from  its  course  by  touching  a  glass  rod. — Experiments  quoted  above, 
on  relative  cohesion. 

Of  liquids  to  liquids. — Oil  and  spirits  of  turpentine  spread  over 
the  surface  of  water. 

Of  gases  to  solids. — Air  and  vapor  of  water  adhere  with  consider- 


Lecture- Notes  on  Physics.  87 

able  force  to  the  surface  of  glass.  This  shown  by  placing  a  beaker 
of  water  under  the  receiver  of  an  air  pump,  when  bubbles  of  air, 
previously  coating  as  a  film  the  surface  of  the  glass,  collect  on  its 
surface.  That  vapor  of  water  also  coats  with  a  film  the  glass,  is 
known  from  the  increase  in  weight  of  a  dry  light  glass  vessel,  when 
exposed  to  a  damp  atmosphere. — The  action  of  clean  platinum  and 
gold  in  condensing  gases  on  their  surfaces. — Charcoal  absorbs  98 
times  its  volume  of  ammonia,  and  14  volumes  of  carbonic  acid  gas. 
As  ammonia  is  condensed  into  a  liquid  by  a  pressure  of  seven  at- 
mospheres, at  a  temperature  of  60°  F.,  it  follows  that  the  absorbed 
gas  must  exist  in  the  liquid  state  in  the  interstices  of  the  charcoal. 
Gold  leaf  will  not  sink  in  water  from  the  air  condensed  on  its  surface. 

Of  gases  to  liquids. — Air  and  all  gas  absorbed  by  water.  Oxygen 
gas  absorbed  from  the  air  by  melted  silver. 

Of  gases  to  gases. — In  the  diffusion  of  gases,  one  gas  acts  as  a 
vacuum  to  another.  Yapor  diffuses  in  space  containing  a  gas,  until 
the  same  tension  is  produced  as  would  have  been  acquired  by  the 
same  vapor  evaporating  from  its  liquid  in  a  vacuum. 

Molecular  Repulsion. 

"When  the  convex  surface  of  a  plano-convex  lens,  of  a  radius  of 
curvature  of  20  feet  or  more,  is  pressed  upon  a  plate  of  glass,  a 
system  of  concentric  colored  rings  are  observed.  These  rings  are 
produced  by  the  interference  of  certain  of  the  rays  of  light  reflected 
from  the  under  surface  of  the  lens  with  those  reflected  from  the 
upper  surface  of  the  glass  plate.  By  knowing  the  diameter  of  any 
ring,  and  the  radius  of  curvature  of  the  lens,  we  can  calculate  the 
distance  between  the  convex  surface  and  the  glass  plate  correspond- 
ing to  the  ring.  Newton  thus  found  that  the  distance  at  each  ring 
exceeded  the  distance  of  the  ring  immediately  within  it  by  the 

ffvl™*11  of  an  incn- 

Now,  unless  the  lens  be  heavy,  or  pressed  against  the  glass  plate, 
no  colored  spot  appears  in  the  centre,  and  it  can  be  shown  that  the 
glasses  are,  in  this  case,  not  in  contact,  but  distant  from  each  other 
at  least  ^^th  of  an  inch,  and  at  this  distance  reposes  the  upper 
glass,  kept  from  the  plate  by  a  repulsion  existing  between  them. 

By  forcing  the  glasses  nearer  together,  we  at  length  produce  a 
Hack  spot  at  the  centre  of  the  ring  system,  and  Prof.  Kobison  found 
that  u  a  very  considerable  force  is  necessary  for  producing  the  black 
spot.  A  greater  pressure  makes  it  broader,  and  in  all  probability 


88  Lecture-Notes  on  Physics. 

this  is  partly  by  the  mutual  yielding  of  the  glasses.  I  found  that 
before  a  spot,  whose  surface  is  a  square  inch,  can  be  produced,  a 
force  exceeding  1,000  pounds  must  be  employed.  When  the  ex- 
periment is  made  with  thin  glasses,  they  are  often  broken  before 
any  black  spot  is  produced. 

"  What  is  it  that  we  properly,  and  without  any  figure  of  speech, 
call  a  pressure  ?  It  is  something  that  we  are  informed  of  solely  by 
our  sense  of  touch.  What  do  we  feel  by  means  of  this  sense,  when 
the  upper  lens  lies  in  our  hand  ?  It  is  not  the  matter  of  this  lens, 
for  we  now  see  that  there  is  some  measurable  distance  between  the 
lens  and  the  hand ;  it  is  this  repulsion.  Give  a  blind  man  a  strong 
magnet  in  his  hand,  and  let  another  person  approach  the  north  pole 
of  a  similar  magnet  to  its  north  pole.  The  blind  man  will  think 
that  the  other  has  pushed  away  the  magnet  he  holds  in  his  hand 
with  something  that  is  soft. 

###          #'».••'     "v*  ^       #..-'  "ii 

"There  is,  therefore,  an  essential  difference  between  mathematical 
and  physical  contact ;  between  the  absolute  annihilation  of  distance 
and  the  actual  pressure  of  adjoining  bodies.  We  must  grant  that 
two  pieces  of  glass  are  not  in  mathematical  contact  till  they  are  ex- 
erting a  mutual  pressure  not  less  than  1,000  pounds  per  inch.  For 
we  must  not  conclude  that  they  are  in  contact  till  the  black  spot 
appears ;  and  even  then  we  dare  not  positively  affirm  it.  My  own 
decided  opinion  is,  that  the  glasses  not  only  are  not  in  mathemati- 
cal contact  in  the  black  spot,  but  the  distance  between  them  is 
vastly  greater  than  the  89,000th  part  of  an  inch,  the  difference  of 
the  distances  at  two  successive  rings. 

•*####*## 

"  While  gravity  produces  sensible  effects  at  the  utmost  boundary 
of  the  solar  system,  these  [attractions  and  repulsions]  seem  limited 
in  their  exertion  to  a  small  fraction  of  an  inch,  perhaps  not  exceed- 
ing ^(juth  part  in  any  instance ;  and  in  this  narrow  bounds  we 
observe  great  diversity  in  the  intensity,  although  we  have  not  yet 
been  able  to  ascertain  the  law  of  variation.  What  is  of  peculiar 
moment,  we  have  seen  that  those  corpuscular  forces  even  change 
their  kind  by  a  change  of  distance,  producing  at  one  distance,  the 
mutual  approach,  and  at  another  distance  the  mutual  separation  of 
the  acting  corpuscles,  from  being  attractive,  becoming  repulsive. 
#  #  *  *  #  *  #  '  * 

"  Physical  contact,  or  pressure,  becomes  sensible  at  the  distance  of 


Lecture- Notes  on  Physics.  89 

the  5000th  part  of  an  inch  nearly,  and  decreases  much  faster  than  in 
the  inverse  duplicate  ratio  of  the  distances.  I  could  infer  this  from 
my  experiments  with  the  glasses  with  great  confidence,  although  I 
could  not  assign  the  precise  law,"  JRofasorts  Mechanical  Philosophy, 
Yol  I.  p.  250,  et  seq. 

All  bodies  expand  when  relieved  of  pressure,  and  this  expansion 
is  caused  by  the  mutual  repulsion  of  their  atoms. 

A  dew  drop  does  not  touch  the  leaf  above  which  it  reposes,  but 
is  held  at  a  certain  distance  by  repulsion.  Certain  insects  walk  on 
water,  which  is  repelled  by  their  feet,  so  that  each  foot  rests  in  a 
pit.  A  needle  floats  on  the  surface  of  water,  in  which  it  forms  a 
trough  in  which  it  rests. 

Prof.  Baden  Powell  has  shown  (Phil  Trans.  1834:,  p.  485),  that 
the  colored  rings  known  as  Newton's  rings,  change  their  breadth 
and  position  in  such  a  manner,  when  the  glasses  which  produce 
them  are  heated,  that  he  inferred  that  the  glasses  repelled  each 
other. 

"  The  distance  at  which  the  repulsive  power  can  act,  is  shown  by 
these  experiments  to  extend  beyond  that  at  which  the  most  extreme 
visible  order  of  Newton's  tints  is  formed.  But  I  have  also  re- 
peated the  experiment  successfully  with  the  colors  formed  under 
the  base  of  a  prism  placed  upon  a  lens  of  a  very  small  convexity ; 
and  according  to  the  analysis  of  these  colors  given  by  Sir  John 
Herschel,  the  distance  is  here  about  the  1100th  of  an  inch." 

Yery  finely  divided  solids,  such  as  elutriated  silica  and  wood- 
ashes  will,  when  rendered  incandescent,  flow  like  liquids  about  the 
capsule  which  contains  them ;  while  it  can  be  directly  shown  that 
a  sensible  distance  exists  between  a  layer  of  these  powders  and  a 
heated  plate  on  which  it  rests. 

The  Molecular  Constitution  of  Matter. — The  Atomic  Theory  of 

Boscovich. 

Of  the  foregoing  facts  which  we  have  brought  together,  concern- 
ing the  general  properties  of  matter  and  the  effects  of  attraction 
and  repulsion  existing  between  the  minute  parts  of  bodies,  we  can 
frame  a  hypothetical  theory  which,  in  a  few  lines,  or  postulates,  will 
embrace  what  we  otherwise  could  not  express  in  many  pages. 

This  theory  together  with  the  doctrine  of  the  Conservation  of 
Energy,  are  the  two  most  important  generalizations  in  Physics,  and, 
12 


90  Lecture-Notes  on  Physics. 

in  our  opinion,  the  following  generalization  forms  an  absolutely 
essential  introduction  to  the  proper  study  of  this  department  of 
science. 

The  ancient  Greek  philosopher,  Democritus,  propounded  an  hypo- 
thesis of  the  constitution  of  matter,  and  gave  the  name  of  atoms  to 
the  ultimate  unalterable  parts  of  which  he  imagined  all  bodies  to 
be  constructed.  In  the  17th  century,  Grassendi  revived  this  hypo- 
thesis, and  attempted  to  develope  it,  while  Newton  used  it  with 
marked  success  in  his  reasonings  on  physical  phenomena ;  but  the 
first  who  formed  a  body  of  doctrine  which  would  embrace  all  known 
facts  in  the  constitution  of  matter,  was  Eoger  Joseph  Boscovich,  of 
Italy,  who  published  at  Vienna,  in  1759,  a  most  important  and  inge- 
nious work,  styled  Theoria  Philosophise  Naturalis  ad  unicam  legem 
virium,  in  Natura  existentium  redacta.  This  is  one  of  the  most  pro- 
found contributions  ever  made  to  science ;  filled  with  curious  and 
important  information,  and  is  well  worthy  of  the  attentive  perusal  of 
the  modern  student.  In  more  recent  days,  the  theory  of  Boscovich 
has  received  further  confirmation  and  extension  in  the  researches  of 
Dalton,  Joule,  Thonison,  Faraday,  Tyndall  and  others. 

We  present  here  a  generalization  which,  while  giving  the  sub- 
stance of  the  important  postulates  of  Boscovich  embraces  others 
made  necessary  by  the  progress  of  science  since  1760. 

1.  Matter  has  trilineal  extension. 

2.  Is  impenetrable. 

3.  Does  not  form  a  plenum. 

4.  All  matter  consists  of  indefinitely  small  but  finite  parts;  of  ex- 
treme hardness ;  indivisible,  and  unalterable  by  either  mechanical 
or  chemical  means ;  and  endued  with  impenetrability  and  inertia. 
These  ultimate  parts  are  called  Atoms. 

5.  These  atoms  are  not  in  mathematical  contact,  but  are  separated 
from  each  other  by  distances  which  are  great  when  compared  with 
the  size  of  the  atoms. 

6.  A  union  of  atoms  forms  a  molecule,  and  combinations  of  mole- 
cules form  particles  of  which  all  bodies  are  composed. 

7.  There  exist  between  the  atoms,  attractions  and  repulsions: 
when  these  tendencies  are  equal,  the  atoms  preserve  fixed  positions 
and  the  volume  of  the  body  is  constant.     These  molecular  forces 
vary  both  in  intensity  and  direction,  by  a  change  of  distance,  so 
that  at  one  distance  two  atoms  attract  each  other,  and  at  another 
they  repel ;  there  being,  within  the  distance  in  which  physical  con 


VolLV,  3*  Scries.Pl.W 


Fig.  3. 


H 


Fig.  4 


Fig.  5. 


Fig.  6. 


Fig.  8. 


.  7. 


Lecture-Notes  on  Physics.  91 

tact  is  observed  (about  5^3 3 th  inch),  several  alternations  of  attrac- 
tion and  repulsion. 

8.  The  repulsion  of  two  atoms  generally  diminishes  more  rapidly 
than  their  attraction  when  the  distance  between  them  is  increased ; 
while  their  repulsion  increases  more  rapidly  than  their  attraction 
when  their  distance  is  diminished. 

9.  The  law  of  variation  is  the  same  in  all  atoms.     It  is  therefore 
mutual ;  for  the  distance  of  a  from  b  being  the  same  as  that  of  I 
from  a,  if  a  attract  or  repel  6,  b  must  attract  or  repel  a  with  exactly 
the  same  force. 

10.  At  all  sensible  distances  (i.  e.  beyond  ^^th  inch),  this  mutual 
tendency  is  attraction,  and  varies  inversely  as  the  squares  of  the 
distances.     It  is  known  as  gravitation. 

11.  The  last  force  which  is  exerted  between  two  atoms  as  their 
distance  diminishes,  is  an  insuperable  repulsion,  so  that  no  force 
however  great  can  press  two  atoms  into  mathematical  contact. 

12.  Between  the  molecules  of  gases  continued  repulsion  seems 
to  exist,  so  that  when  relieved  of  exterior  force,  a  gas  expands 
indefinitely. 

Between  the  molecules  of  liquids  exist  attraction  and  repulsion, 
which  maintain  them  at  determinate  distances ;  but  they  have  no 
fixed  axial  direction,  so  that  a  liquid  molecule  will  rotate  around 
any  imaginary  axis  on  the  action  of  the  slightest  force.  Thus 
liquids  have  fluidity,  while  at  the  same  time  they  have  a  small 
range  of  compressibility. 

In  solids  the  molecules  have,  besides  their  mutual  attractions  and 
repulsions,  polarity  or  fixity  of  position  of  their  axes  inter  se,  so  that 
when  a  molecule  in  any  solid  is  turned  around  any  axis,  it  will  return 
to  its  primitive  position  after  a  series  of  decreasing  oscillations. 

Those  of  the  above  postulates  which  refer  to  the  mutual  action 
of  atoms,  can  be  geometrically  expressed  by  means  of  an  exponen- 
tial curve.  See  Plate  IV.,  Fig.  2. 

Let  an  atom  be  at  A,  while  another  is  anywhere  on  the  line  AX. 
Suppose  that  when  placed  at  i  for  example,  that  an  attraction  exists 
between  the  atoms.  The  intensity  of  this  attraction  is  represented 
by  the  length  of  the  line  i  7,  and  we  show  that  mutual  attraction 
exists  by  drawing  the  ordinate  to  the  point  i  above  the  axis  AX.  If 
the  atom  be  supposed  at  m  and  repelled  by  A,  then  the  repulsion 
and  its  intensity  are  expressed  by  drawing  the  ordinate  to  the  point 
w,  below  the  axis  AX.  This  may  be  supposed  to  be  done  for  every 


92  Lecture-Notes  on  Physics. 

point  of  the  length  AG,  which  represents  the  distance  between  the 
glass  plates  in  Huyghens'  experiment,  or  about  the  3^00  inch,  and 
thus  we  will  form  an  exponential  curve. 

As  there  are  several  alternations  of  attractions  and  repulsions,  the 
curve  will  consist  of  various  inflections  lying  alternately  above  and 
below  AX.  The,  last  inflection,  most  distant  from  A,  viz  :  G'  G"  is 
of  such  a  form  that  the  lengths  of  its  ordinates  being  the  recipro- 
cals of  the  squares  of  their  distances  from  A,  it  expresses  the  law 
of  gravitation ;  the  atom  at  G  being  at  the  point  called  the  limit  of 
gravitation,  or  about  5^0  inch  from  A.  AX  will  be  an  asymptote 
to  this  curve,  while  the  inflection  c'  Dn  will  have  AY  for  asymptote; 
for  the  ordinate  expressing  repulsion  increases  beyond  all  limit  when 
the  distance  from  A  is  just  vanishing.  The  intermediate  branches 
of  the  curve  must  be  determined  by  means  of  the  alternations  of 
attraction  and  repulsion,  in  the  experiments  already  described  and 
by  the  aid  of  the  various  phenomena  of  capillarity  and  of  molecular 
physics. 

If  an  atom,  supposed  at  the  point  c',  or  c",  or  &c.,  have  its  dis- 
tance increased  from  A,  it  will,  being  under  the  curve,  be  attracted 
with  an  intensity  represented  by  its  ordinate.  When  set  free  it  will 
move  with  an  increasing  velocity  towards  its  primitive  position  of 
equilibrium,  which  it  will  surpass  on  account  of  its  inertia,  and, 
coming  into  the  sphere  of  repulsion,  it  will  be  repelled  from  A,  and 
thus  oscillate  about  its  point  of  equilibrium.  The  atom  will  there- 
fore eventually  return  to  c',  or  c",  or  &c.  These  positions  are 
called  limits  of  cohesion,  c'  being  designated  as  the  last  limit  of 
cohesion. 

If  an  atom  at  D',  or  D",  or  &c.,  be  moved  ever  so  little  from  its 
position,  it  will  rush  to  an  adjacent  limit  of  cohesion,  either  to  the 
right  or  to  the  left,  according  as  it  was  moved  from  or  towards  A. 
These  points,  D',  D"  &c.,  are  called  limits  of  dissolution,  and  differ 
from  the  limits  of  cohesion  in  being  positions  of  unstable  equilib- 
rium, and  therefore  only  a  temporary  molecule  can  be  formed  by  an 
atom  placed  at  D',  D",  &c.,  with  an  atom  at  A ;  while  an  atom  at  c', 
c",  &c.,  together  with  the  atom  A  forms  a  permanent  molecule,  which 
resists  compression  and  dilatation^  and  whose  component  atoms 
return  to  their  primitive  positions  when  the  extraneous  force  is 
removed ;  provided  the  compression  or  dilatation  has  not  been  too 
great ;  for,  in  that  case,  the  atom  G",  for  example,  might  be  forced 
beyond  D'  by  a  compression,  or  removed  beyond  D"  by  a  dilatation, 


Lecture-Notes  on  Physics.  93 

and  would  then  rush  to  another  position  of  permanent  equilibrium, 
either  to  c'  or  to  G.  The  only  molecule  that  cannot  possibly  be 
changed  by  compression  is  AC'.  "When,  however,  the  amount  of 
compression  or  dilatation  of  a  body  formed  of  permanent  molecules 
is  a  very  small  fraction  of  its  volume,  the  body  regains  the  dimen- 
sion it  had  before  the  compression  or  dilatation  was  applied,  and  it 
is  found  that  the  compression  or  dilatation  is  proportional  to  the 
force  employed ;  for,  in  this  case,  the  small  portion  of  the  curve 
which  expresses  the  variation  of  repulsion  or  attraction  may  be 
considered  a  straight  line,  and  therefore  its  ordinates  are  as  its 
abscissas. 

These  logical  consequences  of  the  theory  are  confirmed  by  the 
most  extensive  experience.  "  Mr.  Coulomb  was  engaged  (for  a  par- 
ticular purpose),  in  a  series  of  experiments  on  the  oscillations  of 
springs,  particularly  of  twisted  wires.  He  suspended  a  nicely  turned 
ball  or  cylinder  by  a  wire  of  a  certain  length,  and  fitted  it  with  an 
index,  which  pointed  out  the  degrees  of  the  torsion.  He  found  that 
when  a  wire  of  twenty  inches  long  was  twisted  ten  times,  the  index 
returned  to  its  primitive  position,  if  repeated  a  thousand  times,  and 
the  oscillations  were  made  in  equal  times,  whether  wide  or  narrow. 
But  if  it  was  twisted  eleven  times,  the  index  did  not  return  to  its 
first  place,  but  wanted  nearly  a  whole  turn  of  it.  Here,  then,  the 
parts  of  the  wire  had  taken  new  relative  positions,  in  which  they 
were  again  at  rest  But  what  was  most  remarkable  in  Coulomb's 
experiments  was  this:  He  found  that  after  the  wire  had  taken  this 
set  (as  it  is  termed  by  the  artizans),  it  exhibited  the  same  elasticity 
as  before.  It  allowed  a  torsion  of  ten  turns,  and  when  let  go,  it 
returned,  and  after  its  oscillations  were  finished,  it  rested  in  the 
position  from  which  it  had  been  taken.  I  was  much  struck  with 
this  experiment,  and  immediately  repeated  it  on  a  great  variety  of 
substances  with  the  same  result.  The  most  inelastic  substance  that  I 
know  is  soft  clay.  I  got  a  thread  made  of  fine  clay  at  a  pottery, 
by  forcing  it  through  a  syringe.  It  was  about  TJ2th  of  an  inch  in 
diameter,  and  eleven  feet  long.  While  quite  soft  (and  smeared  with 
olive  oil,  to  prevent  its  stiffening  by  the  evaporation  of  its  moisture), 
I  fastened  it  to  the  ceiling,  and  fixed  a  small  weight  and  an  index 
to  its  lower  end.  I  found  that  it  made  5|  turns  a  hundred  times 
and  more,  without  the  smallest  diminution  of  its  elasticity,  always 
recovering  its  first  position.  But  when  I  gave  it  7  turns  it  re- 
turned only  5J.  Thus  it  took  a  set.  In  this  new  arrangement  of 


94  Lecture- Notes  on  Physics. 

its  parts,  I  found  that  it  again  bore  a  twist  of  5J  turns  without 
taking  any  new  set.  And  I  repeated  this  several  times.  I  then 
gave  it  10  turns,  in  the  same  direction  with  the  first  seven.  It 
returned  5J  as  before,  and  was  again  perfectly  elastic  within  this 

limit." 
#•&##  *  •*  ** 

"  Another  appearance  of  tangible  matter  shows  a  most  encour- 
aging conformity  to  the  theory.  Where  bodies  are  very  moderately 
compressed  or  dilated,  the  forces  employed  are  proportional  to  the 
change  of  distance  between  the  particles.  This  appears  most  ex- 
actly true  in  the  experiments  of  Dr.  Hooke,  on  which  he  founded 
his  theory  of  springs,  expressed  in  the  phrase  ut  tensio  sic  vis,  and 
his  noble  improvement  of  pocket  watches  by  applying  a  spiral 
spring  to  the  axis  of  the  balance,  which,  by  its  bending  and  unbend- 
ing, produced  a  force  proportional  to  the  angle  of  the  oscillations, 
and  therefore  made  them  isochronous,  whether  wide  or  narrow.  It 
is  also  confirmed  by  the  experiments  of  Coulomb  on  twisted  wires, 
and  by  the  form  of  the  elastic  curve,  as  determined  by  Bemouilli, 
on  the  supposition  that  the  forces  with  which  the  particles  attracted 
and  repelled  each  other,  are  proportional  to  their  removal  from  their 
natural  quiescent  positions.  But  it  is  found  that  when  the  com- 
pression or  dilatation  is  too  much  increased,  the  resistance  does  not 
increase  so  fast ;  that  it  comes  to  a  maximum  by  still  increasing 
the  strain,  then  decreases,  and  the  body  takes  a  great  set  or  breaks. 
All  this  is  perfectly  analogous  to  the  forces  expressed  by  the  ordi- 
nates  of  our  exponential  curve.  In  the  immediate  vicinity  of  the 
limits  of  cohesion,  the  ordinates  increase  nearly  in  the  ratio  of  the 
abscissae,  then  they  increase  more  slowly,  come  to  a  maximum, 
decrease  again,  till  we  come  to  a  limit  of  dissolution." 


§  VI.   Capillary  Attraction.     " 

The  phenomena  of  capillary  attraction  consist  in  the  elevation 
or  depression  of  the  surfaces  of  liquids  along  the  line  of  contact  with 
the  walls  of  the  vessels  which  contain  them  ;  in  the  ascent  or  depres- 
sion of  liquids  between  slightly  separated  plates,  or  in  tubes  of  such 
small  internal  diameters  as  to  approach  to  the  dimensions  of  a  hair ; 
whence  the  name  of  capillarity,  from  capillus,  a  hair. 

These  effects  are  due  to  the  attractions  of  the  molecules  of  the 


Lecture-Notes  on  Physics. 


95 


liquid  for  each  other  combined  with  the  attractions  existing  between 
them  and  the  molecules  of  the  solid. 

Generalization  of  the  Phenomena  of  Capillary  Attraction. 

1.  The  ascent  or  depression  of  the  liquid  is  inversely  as  the 
diameter  of  the  tube ;  provided  that  this  diameter  does  not  exceed 
two  millimetres.     In  tubes  over  twenty  millimetres  in  diameter, 
there  is  neither  elevation  or  depression  of  liquids. 

2.  The  phenomena  are  independent  of  the  pressure  to  which  the 
apparatus  is  subjected ;  being  the  same  in  vacuo  as  in  compressed 
air. 

3.  They  do  not  depend  on  the  thickness  of  the  tube ;  hence  the 
action  of  the  tube  is  limited  in  its  effects  to  insensible  distances. 

4.  The  phenomena  vary  with  the  material  of  the  tube,  and  with 
the  nature  of  the  liquid ;  thus,  in  a  tube  of  glass,  water  rises  above 
and  mercury  is  depressed  below  the  level  of  the  outside  liquid.   The 
following  table  of  the  experiments  of  M.  Frankenheirn,  gives  the 
heights  in  millimetres  to  which  different  liquids  rise,  at  a  tempera- 
ture of  0°  C.,  in  a  glass  tube  of  1  millimetre  in  diameter. 


LIQUIDS. 

DENSITY. 

ELEVATION. 

Water  „  

1-000 

30-73 

Formic  Acid 

1-105 

20-40 

Acetic  Acid     ..  . 

1-290 

17-02 

Sulphuric  Acid  

1-840 

16  80 

Solution  of  Potassa.  .  . 
Petroleum  

1-274 

0-847 

15-40 
13-90 

Spirits  of  Turpentine.  . 

0890 
0-905 

13-52 
12-20 

Alcohol          .       

0-821 

12-10 

Alcohol                  ... 

0-967 

14-54 

Ether  

0  737 

10-80 

Bisulphide  of  Carbon.. 

1-290 

10-20 

5.  When  the  liquid  wets  the  tube,  it  rises  above  the  level  of  the 


96  Lecture- Notes  on  Physics. 

liquid  outside  the  tube ;  and  in  this  case  the  surface  of  the  elevated 
liquid  is  concave. 

Example.     "Water  in  glass  tube. 

6.  When  the  liquid  does  not  wet  the  tube,  it  is  depressed  below  the 
surface  of  the  exterior  liquid ;  and  in  this  case,  the  surface  of  the 
liquid  in  the  tube  is  convex. 

Example.     Mercury  in  tube  of  glass. 

7.  When  the  liquid  in  the  tube  has  a  plane  surface,  there  is  neither 
elevation  or  depression. 

Example,     Water  in  a  tube  of  steel. 

These  facts  are  readily  explained  by  the  atomic  theory,,  of  which 
they  are  a  beautiful  illustration  and  a  natural  deduction. 

(a.)  An  attraction  exists  between  the  neighboring  molecules  of  a 
liquid,  and  between  the  molecules  of  a  liquid  and  of  the  contiguous 
solid. 

(b.)  This  force  decreases  very  rapidly  as  the  distance  between  the 
molecules  increases,  and  becomes  null  when  that  distance  exceeds 
the  radius  of  sensible  attraction. 

(c.)  The  attraction  existing  between  the  molecules  forming  the 
surface  of  a  liquid,  and  those  extending  below  the  surface  as  far  as 
the  radius  of  sensible  attraction,  produces  a  molecular  pressure,  or 
tension,  on  this  surface,  whose  effect  has  to  be  added  to  the  pressures 
produced  by  gravity  and  the  atmosphere. 

(d.)  The  molecular  pressure  is  greater  with  a  convex  and  less 
with  a  concave  than  with  a  plane  liquid  surface. 

The  truth  of  the  four  preceding  postulates,  is  made  clear  by  what 
follows : 

Let  s  s',  Fig,  3,  be  a  liquid  surface  of  any  form.  M  is  a  molecule 
on  the  surface ;  Mr  is  a  molecule  distant  from  the  surface  less  than 
the  radius  of  sensible  attraction ;  and  M"  a  molecule  whose  dis- 
tance from  the  surface  equals  the  radius  of  sensible  attraction; 
while  all  molecules  between  s  s'  and  R  R'  are  distant  from  the  sur- 
face less  than  the  radius  of  sensible  attraction. 

The  molecule,  M,  on  the  surface,  is  attracted  downward  by  all 
the  molecules  contained  in  the  portion  of  sphere  which  has  for  its 
radius  M  P,  the  radius  of  sensible  attraction.  The  effect  of  all  these 
attractions  on  M  will  be  a  resultant  in  the  direction  M  P,  perpen- 
dicular to  the  surface. 

j      The  molecule,  M',  is  attracted  by  all  the  molecules  contained  in 
he  spherical  portion  ABC,  which  we  can  divide  into  three  parts  by 


Lecture- Notes  on  Physics.  97 

three  equidistant  planes,  A  B,  p  Q,  A'  B',  paralleFto  the  surface,  s  s'. 
The  attraction  produced  by  A  B  p  Q,  is  destroyed  by  the  attraction  of 
p  Q  A'  B',  and  therefore  the  molecule  M7  is  drawn  downward  as 
though  it  were  attracted  only  by  the  liquid  contained  in  A'  B'  c, 
which  gives  a  resultant,  p',  also  perpendicular  to  the  surface,  but 
less  than  P. 

The  molecule,  M",  whose  distance  from  the  surface  equals  the 
radius  of  sensible  attraction,  and  all  other  molecules  placed  at 
greater  distances,  are  equally  attracted  on  all  sides,  and  therefore 
they  produce  no  tension  in  the  surface-film  of  the  liquid,  which  has 
for  its  thickness  the  radius  of  sensible  attraction. 

The  influence  of  the  curvature  of  the  liquid  surface  on  the  molecular 
pressure. 

Let  M',  Fig.  4,  be  a  molecule  at  a  distance  M  H  from  the  surface 
s  s'  of  the  liquid.  With  M  as  a  centre,  draw  a  sphere  whose  radius 
M'  P  equals  the  radius  of  sensible  attraction. 

If  the  surface  is  a  plane,  AB,  the  attractions  of  the  liquid  in 
A  B  P  Q  are  destroyed  by  those  produced  by  the  symmetrical  portion 
below,  A'  Br  p  Q,  and  there  remains  for  resultant  only  the  action  of 
A'  B'  c. 

•  Suppose  the  surface  concave  and  D  H  E ;  if  we  draw  through  H' 
the  symmetrical  surface,  D'  H'  E',  it  is  evident  that  the  attractions 
of  the  molecules  comprised  between  D  H  E  P  Q  and  of  those  contained 
within  D'  H'  E'  P  Q  equal  and  oppose  each  other,  and  there  remains 
only  the  attraction  of  D'  H'  E'  c  on  M',  which  is  less  than  when  the 
surface  was  a  plane. 

If  the  surface  is  convex,  and  is  represented  by  K  H  L,  draw  the 
symmetrical  surface,  K'  H'  I/ ;  then  the  efficient  attracting  portion 
of  the  liquid  will  be  increased  and  represented  by  Kr  G'  I/,  and  con- 
sequently the  molecular  pressure  is  greater  with  a  convex  than  with 
a  plane  surface. 

We  can  now  explain  the  rise  and  depression  of  liquids  in  capillary 
tubes. 

When  the  surface  of  the  liquid  in  the  tube  is  concave,  the  mole- 
cular pressure  on  the  liquid  in  the  tube  is  less  than  the  pressure  on 
the  liquid  outside  the  tube,  and  therefore  the  liquid  rises  in  the  tube 
to  a  height  which  measures  the  diminution  of  pressure  produced 
by  the  concave  surface. 

When  the  surface  of  the  liquid  in  the  tube  is  plane,  there  is 
13 


98  Lecture- Notes  on  Physics. 

neither  elevation  or  depression,  for  the  pressures  are  the  same  on 
the  surfaces  of  the  liquid  inside  and  outside  the  tube. 

When  the  surface  of  the  liquid  in  the  tube  is  convex,  the  mole- 
cular pressure  on  the  liquid  in  the  tube  is  more  than  the  pressure 
on  the  liquid  outside  the  tube,  and  therefore  the  liquid  column  is 
depressed  in  the  tube  below  the  level  of  the  outside  liquid,  and  the 
depth  to  which  the  column  is  forced  below  this  level,  is  the  measure 
of  the  pressure  produced  by  the  convex  surface. 

As  we  have  seen  that  the  elevation  or  depression  of  liquids  in 
capillary  tubes,  is  due  to  a  diminution  or  increase  of  molecular  pres- 
sure, produced  by  a  concave  or  convex  surface,  it  remains,  to  ren- 
der the  explanation  complete,  to  show  the  cause  of  the  special  figure 
of  each  surface. 

Cause  of  the  (1)  plane,  (2)  concave,  and  (3)  convex  surfaces  of 
liquids  in  capillary  tubes. 

Let  D  A  in  Figs.  5,  6  and  7,  be  the  vertical  surface  of  a  solid 
plunged  in  liquids,  whose  surfaces  are  M  L.  Let  M  be  a  molecule  of 
the  surface  of  the  liquid  contiguous  to  the  plate.  This  molecule  is 
attracted  by  all  the  molecules  contained  in  the  quarter-spheres 
D  M  c  and  A  M  c,  whose  radii  are  equal  to  the  distance  of  sensible 
attraction ;  giving  as  resultants  M  s  and  M  s',  while  the  resultant  of 
the  attractions  of  the  liquid  on  the  molecule,  M,  will  be  M.  p. 

Three  cases  can  present  themselves. 

1.  If  the  resultant,  M  P,  Fig.  5,  is  twice  M  s,  or  its  equal,  M  s',  the 
effect  of  these  three  attractions  on  M  will  be  the  resultant,  M  K  ; 
which  being  perpendicular  to  the   liquid  surface,  the  fluid  will 
remain  horizontal,  for  the  surface  of  a  liquid  is  always  perpendicular 
to  the  forces  acting  on  it. 

2.  If  the  resultant,  M.  P,  Fig.  6,  is  less  than  twice  M  s  or  M  s',  the 
three  attractions  will  result  in  M  K,  which  will  produce  a  concave 
surface  M  I/,  inclined  against  the  plate. 

3.  If  the  resultant,  M  P,  Fig.  7,  is  more  than  twice  M  s  or  M  s',  the 
resultant  of  the  three  attractions  on  liquid  contiguous  to  solid  will 
be  M  R,  which  will,  for  the  reason  given  above,  produce  the  surface 
Mr  L',  which  will  be  convex. 

The  above  results  may  be  expressed  concisely  as  follows : 

I.  On  the  free  surface  of  every  liquid  there  exists  a  molecular 
pressure  from  without  inward,  which  always  adds  its  effect  to  that 
produced  by  gravity  and  the  pressure  of  the  air. 

II.  The  intensity  of  this  molecular  pressure  varies  with  the  form 


Lecture-Notes  on  Physics.  99 

of  the  surface,  being  greater  when  the  surface  is  convex  and  less 
when  concave,  than  when  it  is  plane. 

III.  The  form  of  a  liquid  surface  in  a  tube,  depends  on  the  rela- 
tive amounts  of  attraction  existing  between  the  molecules  of  the 
liquid  and  the  molecules  of  the  solid  and  of  the  liquid. 

1.  When  the  attraction  between  the  molecules  of  the  liquid  is 
twice  as  great  as  the  attraction  between  the  molecules  of  the  liquid 
and  those  contained  in  an  equal  portion  of  the  solid,  the  surface  in 
the  capillary  tube  is  horizontal. 

2.  When  the  attraction  between  the  molecules  of  the  liquid  is 
less  than  twice  that  existing  between  the  molecules  of  the  liquid  and 
solid,  the  surface  in  the  tube  is  concave. 

3.  When  the  attraction  between  the  molecules  of  the  liquid  is 
more  than  twice  that  between  the  molecules  of  the  liquid  and  solid 
the  surface  in  the  tube  is  convex. 

IV.  When  the  surface  of  the  liquid  in  the  capillary  tube  is 
(a)  horizontal,  it  is  in  the  same  plane  with  the  exterior  liquid. 
(/;)  concave,  it  is  above  the  plane  of  the  exterior  liquid,     (c)  convex, 
it  is  below  the  plane  of  the  exterior  liquid. 

V.  The  amount  of  elevation  or  of  depression  of  the  same  liquid 
in  tubes  of  the  same  material,  is  inversely  as  the  diameter  of  these 
tubes.     This  is  known  as  the  law  of  Jurin,  after  the  philosopher 
who  established  it ;  and  with  the  aid  of  the  table  already  given 
we  can  by  means  of  it  readily  calculate  the  heights  to  which  different 
liquids  will  rise  in  glass  tubes  of  various  dimensions,  contained 
within  diameters  of  two   millimetres   to  a  few  hundredths   of  a 
millimetre. 

The  reason  of  this  law  is  as  follows.  The  force  which  elevates  or 
depresses  the  liquid  columns  in  the  tubes  depends  evidently,  from 
what  has  preceded,  upon  the  number  of  the  molecules  on  the  sur- 
face of  the  liquid  contiguous  to  the  sides  of  the  tubes.  Therefore, 
the  forces  of  elevation  or  of  depression  are  as  the  interior  circum- 
ferences of  the  tubes,  arid  the  forces  are  measured  by  the  quantity 
(or  weight)  of  liquid  elevated  above  or  depressed  below  the  level  of 
the  liquid  exterior  to  the  capillary  tube.  Therefore,  let  h  and  hf 
be  the  lengths  of  liquid  columns  elevated  or  depressed  in  tubes 
whose  interior  diameters  are  respectively  d  and  a" '.  Their  interior 
circumferences  are  ft  d  and  it  df .  6  being  the  specific  gravity  of  the 
liquid,  the  weights  of  the  columns  elevated  or  depressed  will  be 
Jrtc?2/i5,  and  JrfcTVi'S.  These  weights  are  equal  to  the  force* 


100  Lecture-Notes  on  Physics. 

which  produce  the  elevations  or  depressions  of  the  liquid  columns, 
and  these  forces  being  to  each  other  as  the  interior  circumferences 
of  the  tubes,  we  have 


or 

d:d'::h':h 
which  is  the  expression  of  the  law  given  above. 

Experiments.  —  The  apparatus  with  which  Gay  Lussac  verified 
the  above  law,  explained  and  used. 

If  two  squares  of  plane  glass,  touching  along  two  vertical  edges, 
are  opened  to  an  acute  angle  and  placed  in  colored  water,  the  liquid 
will  rise  between  the  plates,  forming  an  equilateral  hyperbola,  and 
therefore  the  liquid  at  various  points  stands  at  heights  inversely  as 
the  distance  of  the  plates  at  these  points. 

The  relation  which  exists  between  the  form  of  the  surface  which 
terminates  the  capillary  column  and  its  vertical  distance  above  the 
plane  of  the  exterior  liquid,  is  beautifully  shown  by  the  following 
experiment,  which,  with  those  above  cited,  can  be  readily  thrown 
on  a  screen  by  means  of  the  lantern  and  erecting  prism  of  Prof. 
Morton  (see  Journal  of  Franklin  Institute,  Yol.  LIIL,  p.  406).  A 
large  glass  tube  has  connected  with  it  a  capillary  tube,  as  shown  in 
Fig.  8.  Water,  colored  with  carmine,  is  poured  into  the  larger 
tube  until  its  level  reaches,  say,  A,  and  the  liquid  in  the  capillary 
tube  just  attains  the  top,  s,  and  in  these  circumstances,  will  there 
form  a  concave  surface.  Now,  on  pouring  into  the  large  tube  more 
liquid,  the  concave  surface  becomes  flatter  and  flatter  as  the  liquid 
rises  in  the  tube  A,  until,  when  the  surface  rises  to  B  on  the  same 
level  as  s,  the  terminal  surface  at  s  is  a  plane.  When  liquid  is 
further  added  until  the  surface  reaches  c,  at  a  higher  level  than  s, 
the  capillary  surface  at  s  is  convex. 

When  s  is  concave,  the  molecular  pressure  on  this  surface  is  less 
than  on  A  by  the  pressure  of  the  column  from  the  level  A  to  s. 
When  s  is  plane,  equality  of  pressure  exists  in  both  tubes,  and 
therefore  the  liquid  surfaces  are  in  the  same  plane.  When  s  is 
convex,  more  molecular  pressure  is  on  s  than  on  c,  by  the  column 
from  s  to  the  level  c, 

Professor  Plateau,  of  the  University  of  Ghent,  has  made  a  series 
of  very  important  investigations  in  molecular  physics,  which  are 
contained  in  a  series  of  papers  entitled,  "  Experimental  and  Theo- 
retical Researches  on  the  Figures  of  Equilibrium,  of  a  Liquid  Mass 


Lecture-Notes  on  Physics.  101 

withdrawn  from  the  action  of  Gravity"  translated  and  published  by 
the  Smithsonian  Institution,  in  the  Reports  of  1863,  et  seq.  The 
fifth  series  of  these  investigations  (Smith.  Rep.  1865),  contains  a 
research  on  the  molecular  pressure  exerted  by  liquid  films,  with 
applications  to  capillary  action ;  and  so  interesting  has  this  investi- 
gation appeared  to  us,  that  we  thought  it  proper  to  present  a  rather 
full  abstract  from  Prof.  Plateau's  paper. 

Pressure  exerted  by  a  spherical  film,  on  the  air  which  it  contains. 
— App  lication. 

11  The  exterior  surface  of  a  laminar  sphere  being  convex  in  every 
direction,  the  pressure  which  corresponds  to  it  is  greater  than  that 
of  a  plane  surface,  and  consequently  the  resultant  of  the  pressures 
exerted  in  any  point  of  the  bubble  by  the  two  surfaces  of  the  latter, 
is  directed  towards  the  interior ;  whence  it  results  that  the  bubble 
presses  on  the  air  which  it  encloses.  It  is,  indeed,  well  known 
that  when  a  soap  bubble  has  been  inflated,  and  while  it  is  still 
attached  to  the  tube,  if  the  other  extremity  of  this  last  be  left  open, 
the  bubble  gradually  collapses,  expelling  the  air  which  it  contained 
through  the  tube.  We  see  now  what  is  the  precise  cause  of  this 
expulsion. 

"  But  we  may  go  further,  and  determine  according  to  what  law  it 
is,  that  the  pressure,  exerted  by  such  a  bubble  on  the  confined  air, 
depends  on  the  diameter  of  that  bubble.  We  can  compute,  more- 
over, the  exact  value  of  the  pressure  in  question  for  a  bubble  hav- 
ing a  given  diameter,  and  formed  of  a  given  liquid.  The  pressure 
corresponding  to  a  point  of  a  laminar  figure,  has  for  its  expression 

A( h      , -).     R  and  R',  standing  for  the  radii  of  curvature,  P 

\    R  R    / 

being  the  pressure  which  a  plane  surface  would  occasion,  and  A  a 
constant  which  depends  on  the  nature  of  the  liquid.  Now,  in  the 
case  of  the  spherical  figure,  we  have  R  =  R/  =  the  radius  of  the 
sphere.  If,  therefore,  we  designate  by  d  the  diameter  of  the  bubble, 

the  value  of  the  pressure  will  simply  become——- ,  always,  be  it  un- 

Cu 

derstood,  neglecting  the  slight  thickness  of  the  film ;  whence  it  fol- 
lows that  the  intensity  of  the  pressure  exerted  by  a  laminar  spherical 
bubble  on  the  air  which.it  confines,  is  in  inverse  ratio  to  the  diameter 
of  that  bubble. 

"This  first  result  established,  let  us  recur  to  the  general  expres- 
sion of  the  pressure  corresponding  to  any  point  of  a  liquid  surface, 


102  Lecture-Notes  on  Physics. 

an  expression  which  is  p  +  -=    I  — }-•-,  ).     For  a  surface  of  convex 

A     \  R      R    / 

spherical  curvature,  if  we  designate  by  d  the  diameter  of  the  sphere 
to  which  this  surface  pertains,  the  above  expression  becomes 

2  A 
P  H-  -7- ,  and  for  a  spherical  surface  of  concave  curvature  pertaining 

2  A 

to  a  sphere  of  the  same  diameter,  we  shall  have  P -,-.     Thus,  in 

ct 

the  case  of  the  convex  surface  the  total  pressure  is  the  sum  of  two 
forces  acting  in  the  same  direction — force,  of  which  one  designated 
by  P  is  the  pressure  which  a  plane  surface  would  exert,  and  the 

2  A 

other  represented  by  -=-  is  the  action  which  depends  on  the  curva- 
ture. On  the  contrary,  in  the  case  of  the  concave  surface  the  total 
pressure  is  the  difference  between  two  forces  acting  in  opposite 
directions,  and  which  are  again,  one  the  action  P  of  a  plane  surface, 

2  A 
and  the  other  —^   which  depends  on  the  curvature.     Whence  it  is 

4  A 

seen  that  the  quantity  —r- ,  which  represents  the  pressure  exerted 

by  a  spherical  film  on  the  air  it  encloses,  is  equal  to  double  the 
action  which  proceeds  from  the  curvature  of  one  or  the  other  sur- 
face of  the  film. 

"Now,  when  a  liquid  rises  in  a  capillary  tube,  and  the  diameter  of 
this  is  sufficiently  small,  we  know  that  the  surface  which  terminates 
the  column  raised  'does  not  differ  sensibly  from  a  concave  hemi- 
sphere, whose  diameter  is  consequently  equal  to  that  of  the  tube. 
Let  us  recall,  moreover,  a  part  of  the  reasoning  by  which  we  arrive, 
in  the  theory  of  capillary  action,  at  the  law  which  connects  the 
height  of  the  column  raised  with  the  diameter  of  the  tube.  Let  us 
suppose  a  pipe,  excessively  slender,  proceeding  from  the  lowest  point 
of  the  hemispheric  surface  in  question,  descending  vertically  to  the 
lower  orifice  of  the  tube,  then  bending  horizontally,  and  finally 
rising  again  so  as  to  terminate  vertically  at  a  point  of  the  plane 
surface  of  the  liquid  exterior  to  the  tube.  The  pressures  corres- 
ponding to  the  two  orifices  of  this  little  pipe  will  be,  on  the  one 

2  A 

part,  P,  and  on  the  other,  p — ,  if  by  5  be  designated  the  diame- 
ter of  the  concave  hemisphere,  or,  what  amounts  to  the  same  thing, 
vthat  of  the  tube.  Now,  the  two  forces  P,  mutually  destroying  one 


Lecture- Notes  on  Physics.  103 

2  A 
another,  there  remains  only  the  force — ,  which,  having  a  sign 

contrary  to  that  of  P,  acts  consequently  from  below  upwards  at  the 
lower  point  of  the  concave  hemisphere,  and  it  is  this  which  sustains 
the  weight  of  the  molecular  thread  contained  in  the  first  branch  of 
the  little  pipe  between  the  point  just  mentioned  and  a  point  situ- 
ated at  the  height  of  the  exterior  level.  This  premised,  let  us  re- 

2  A 
mark  that  the  quantity         is  the  action  which  results  from  the 

curvature  of  the  concave  surface.     The  double  of  this  quantity  or 

4  A 

— -,  will  therefore  express  the  pressure  exerted  on  the  enclosed 

air  by  a  laminar  sphere  or  hollow  bubble  of  the  diameter  5,  and 
formed  of  the  same  liquid.  It  thence  results  that  this  pressure  con- 
stitutes a  force  capable  of  sustaining  the  liquid  at  a  height  double 
that  to  which  it  rises  in  the  capillary  tube,  and  that,  consequently, 
it  would  form  an  equilibrium  to  the  pressure  of  a  column  of  the 
same  liquid  having  that  double  height.  Let  us  suppose,  for  the 
sake  of  precision,  J  equal  to  a  millimetre,  and  designate  by  h  the 
height  at  which  the  liquid  stops  in  a  tube  of  that  diameter.  We 
shall  have  this  new  result,  that  the  pressure  exerted  on  the  enclosed 
air  by  a  hollow  bubble  formed  of  a  given  liquid  and  having  a 
diameter  of  1  millimetre,  would  form  an  equilibrium  to  that  exerted 
by  a  column  of  this  liquid  of  a  height  equal  to  2  h.  Now,  the  pres- 
sure exerted  by  a  bubble  being  in  inverse  ratio  to  the  diameter 
thereof,  it  follows  that  the  liquid  column  which  would  form  an 
equilibrium  to  the  pressure  exerted  by  a  bubble  of  any  diameter 

whatever,  c?,  will  have  a  height  equal  to   ^-. 

"It  would  seem,  at  first,  that  this  last  expression  ought  to  apply 
equally  well  to  liquids  which  sink  in  capillary  txibes,  h  then  desig- 
nating this  subsidence,  the  tube  still  being  supposed  1  millimetre 
in  diameter ;  but  is  not  altogether  so,  for  that  would  require,  as  is 
readily  seen  by  the  reasonings  which  precede,  that  the  surface  which 
terminates  the  depressed  column  in  the  capillary  tube  should  be 
sensibly  a  convex  hemisphere ;  now  we  know  that  in  the  case  of 
mercury  this  surface  is  less  curved ;  according  to  the  observations 
of  M.  Bede,  its  height  is  but  about  half  of  the  radius  of  the  tube ; 
whence  it  follows  that  the  valuation  of  the  pressure  yielded  by  our 
formula  would  be  too  small  in  regard  to  such  liquids.  It  may  be 
considered,  however,  as  a  first  approximation. 


104  Lecture-Notes  on  Physics. 

Let  us  take,  as  a  measure  of  the  pressure  exerted  by  a  bubble, 
the  height  of  the  column  of  water  to  which  it  would  form  an  equili- 
brium. Then,  if  £  designates  the  density  of  the  liquid  of  which  the 
bubble  is  formed,  that  of  water  being  1,  the  heights  of  the  columns 
of  water  and  of  the  liquid  in  question  which  would  form  an 
equilibrium  to  the  same  pressure  will  be  to  one  another,  in  the  in- 
verse ratio  of  the  densities,  and,  therefore,  if  the  height  of  the 

second  is  -^,  that  of  the  first  will  be  --*— .  Hence,  designating 
by  p  the  pressure  exerted  by  a  laminar  sphere  on  the  air  which  it 
encloses,  we  obtain  definitely  p  —  J—^-- ,  ^  being,  as  we  have  seen, 

the  density  of  the  liquid  which  constitutes  the  film,  li  the  height 
to  which  this  liquid  rises  in  a  capillary  tube  1  millimetre  in  dia- 
meter, and  d  the  diameter  of  the  bubble.  If,  for  example,  the  bub- 
ble be  formed  of  pure  water,  we.  have  £  =  1,  and,  according  to 
the  measurements  taken  by  physicists,  we  have,  very  exactly, 
^  =  30mm;  the  above  formula,  therefore,  will  give,  in  this  case, 

60 
p  =  -^-.     If  we  could  form  a  bubble  of  pure  water  of  one  decimetre 

or  100mm,  in  diameter,  the  pressure  which  it  would  exert  would 
consequently  be  equal  to  Omm*6,  or,  in  other  terms,  would  form  an 
equilibrium  to  the  pressure  of  a  column  of  water  Omm'6  in  height; 
the  pressure  exerted  by  a  bubble  of  the  same  liquid  one  centimetre, 
or  10mm  in  diameter,  would  form  an  equilibrium  to  that  of  a  column 
of  water  6mm.  As  regards  soap-bubbles,  their  pressures,  if  the 
solution  were  as  weak  as  possible,  would  differ  very  little  from 
those  exerted  by  bubbles  of  the  same  diameters  formed  of  pure 
water. 

"For  mercury  we  have  £  —  13*59,  and,  according  to  M.  Bede,  h 
about  equal  to  10mm;  the  formula  would  therefore  give,  for  a  bub- 

971  '8 
ble  of  mercury  p  = — -^— ,  but,  from  the  remark  which  closes  the 

last  paragraph,  this  value  is  too  weak,  and  can  only  be  regarded  as 
a  first  approximation.  It  only  instructs  us  that,  with  an  equality 
of  diameter,  the  pressure  of  a  bubble  of  mercury  would  exceed  four 
and  a  half  times  that  of  a  bubble  of  pure  water.  For  sulphuric 
ether,  we  have  £  —  0'715,  and  conclude  from  measurements  taken 
from  M.  Frankenheim,  h  to  be  very  closely  to  10Rim>2  ;  whence  re- 


Lecture- Notes  on  Physics.  105 

suits  j9— — -T-,  and  thus,  with  an  equal  diameter,  the  pressure  of  a 

bubble  of  sulphuric  ether  would  be  but  the  fourth  of  that  of  a  bub- . 
ble  of  pure  water. 

"We  know  that  the  product  h  »  being  the  product  of  the  capillary 
height  by  the  density,  is  proportional  to  the  molecular  attraction 
of  the  liquid  for  itself,  or,  in  other  terms,  to  the  cohesion  of  the 
liquid ;  (see  research  of  Prof.  Henry  on  Cohesion  of  Liquids,  quoted 
in  §Y.)  it  is,  moreover,  the  result  from  a  comparison  of  the  values 

— j-  and  — -j— ,  which  have  been  found  to  represent  the  pressure 

exerted  by  a  laminar  sphere  on  the  air  which  it  contains ;  hence 
we  deduce  h  $  =  2  A,  and  it  will  be  remembered  that  A  is  the  capil- 
lary constant ;  that  is  to  say,  a  quantity  proportional  to  the  cohe- 

2  hs 
sion  of  the  liquid.     The  formula  p  ==  — -=-    indicates,   therefore,   as 

must  be  evident,  that  the  pressure  exerted  by  a  laminary  bubble 
on  the  included  air  is  in  the  direct  ratio  of  the  cohesion  of  the  liquid 
which  constitutes  the  film  and  the  inverse  ratio  of  the  diameter  of 
the  bubble. 

"  As  early  as  1830,  a  learned  American,  Dr.  Hough,  had  sought 
to  arrive  at  the  measure  of  pressure  exerted,  whether  on  a  bubble 
of  air  contained  in  an  indefinite  liquid  or  on  the  air  enclosed  in  a 
bubble  of  soap. 

(Inquiries  into  the  principles  of  liquid  attraction. — Silliman's 
Journal,  1st  series,  vol.  xvii.,  page  86.) 

"  He  conceives  quite  a  just  idea  of  the  cause  of  these  pressures 
which  he  does  not,  however,  distinguish  from  one  another,  and,  in 
order  to  appreciate  them,  sets  out,  as  I  have  done,  with  a  considera- 
tion of  the  concave  surface  which  terminates  a  column  of  the  same 
liquid  raised  in  a  capillary  tube ;  but,  although  an  ingenious  ob- 
server, he  was  deficient  in  a  knowledge  of  the  theory  of  capillary 
action,  and  hence  arrives,  by  reasoning,  of  which  the  error  is  palpa- 
ble, at  values  and  a  law  which  are  necessarily  false. 

"Prof.  Henry,  in  a  very  remarkable  verbal  communication  on  the 
cohesion  of  liquids,  made  in  1844,  to  the  American  Philosophical 
Society,  (Philosophical  Magazine,  1845,  vol.  xxvi.,  page  541),  de- 
scribed experiments  by  means  of  which  he  had  sought  to  measure 
the  pressure  exerted  on  the  internal  air  by  a  bubble  of  soap  of  a 
given  diameter.  According  to  the  account  rendered  of  this  com- 
14 


106  Lecture-Notes  on  Physics. 

munication,  the  mode  of  operation  adopted  by  Mr.  Henry  was  essen- 
tially as  follows :  lie  availed  himself  of  a  glass  tube  of  U  form,  of 
.  small  interior  diameter,  one  of  whose  branches  was  bell-shaped  at 
its  extremity,  and  inflated  a  soap-bubble  extending  to  the  edge  of 
this  widened  portion ;  he  then  introduced  into  the  tube  a  certain 
quantity  of  water,  and  the  difference  of  level  in  the  two  branches 
gave  him  the  measure  of  the  pressure.  Unfortunately,  the  state- 
ment given  does  not  make  known  the  numbers  obtained,  nor  does 
it  appear  that  Mr.  Henry  has  subsequently  published  them.  This 
physicist  refers  the  phenomenon  to  its  real  cause,  and  states  the  law 
which  connects  the  pressure  with  the  diameter  of  the  bubble;  the 
account  does  not  say  whether  the  experiments  verified  it.  But  Mr. 
Henry  considers  that  a  hollow  bubble  may  be  assimilated  to  a  full 
sphere  reduced  to  its  compressing  surface;  that  is  to  say,  he  attri- 
butes the  phenomenon  to  the  action  of  the  exterior  surface  of  the 
bubble,  without  taking  into  account  that  of  the  interior  surface. 
Let  us  add  that,  in  the  same  communication,  Mr.  Henry  has  men- 
tioned several  experiments  which  he  had  made  on  the  films  of  soap 
and  water,  and  which,  from  the  statement  given  would  elucidate, 
in  a  remarkable  manner,  the  principles  of  the  theory  of  capillary 
action.  It  is  much  to  be  regretted  that  these  experiments  are 
not  described 

"  In  a  memoir  presented  to  the  Philomatic  Society  in  1856,  and 
printed  in  1859  in  the  Comptes  Rendus,  (tome  xlviii.,  pap^  "*  405.) 
M.  de  Tessan  maintains  that  if  the  vapor  which  forms  olouc  •  ,*•.'.  /• 
fogs  were  composed  of  vesicles,  the  air  enclosed  in  a  vesicle  of  0*0- 
millimetre  diameter  would  be  subjected,  on  the  part  of  this  vesicle, 
to  a  pressure  equivalent  to  \  of  an  atmosphere.  M.  de  Tessan  does 
not  say  in  what  manner  he  obtained  this  valuation ;  but  it  is  easily 
seen  that  he  has  fallen  into  an  error  analogous  to  that  of  Professor 
Henry,  in  the  sense  that  he  pays  no  attention  except  to  the  exterior 
surface  of  the  liquid  pellicle.  According  to  the  formula  of  the  pre- 
ceding paragraph,  the  pressure  exerted  on  the  interior  air  by  a 
bubble  of  water  of  0*02  millimetre  diameter  would,  in  fact,  be  equi- 
valent to  that  of  a  column  of  water  3  metres  in  height,  which  equals 
nearly  f  of  the  atmospheric  pressure ;  M.  de  Tessan  has  found  then 
but  half  the  real  value,  and  we  know  that  this  half  is  the  action  due 
to  the  curvature  of  one  only  of  the  surfaces  of  the  film. 

"  After  having  obtained  the  general  expression  of  the  pressure 
exerted  by  a  laminar  sphere  on  the  air  which  it  encloses,  it  r  em  aim  d 


Lecture- Notes  on  Physics. 


107 


for  me  to  submit  my  formula  to  the  control  of  experiment.  I  have 
employed,  with  that  view,  the  process  of  Mr.  Henry,  which  means 
that  the  pressure  was  directly  measured  by  the  height  of  the  column 
of  water  to  which  it  formed  an  equilibrium. 

From  our  formula  we  deduce  p  d  =  2  h  $  ;  for  the  same  liquid, 
and  at  the  same  temperature,  the  product  of  the  pressure  by  the 
diameter  of  the  bubble  must,  therefore,  be  constant,  since  h  and  £ 
are  so.  It  is  this  constancy  which  I  have  first  sought  to  verify  for 
bubbles  of  giyceric  liquid — (a  solution  of  Marseilles  soap  and  gly- 
cerine in  distilled  water,  with  which  Plateau  made  his  bubbles) — 
of  different  diameters." 

Plateau  here  describes  his  apparatus  and  the  precautions  to  be 
used  in  making  the  measures,  which  the  reader  will  find  detailed 
in  the  Smithsonian  Report  for  1865. 

"  The  following  table  contains  the  results  of  these  experiments;  I 
have  arranged  them,  not  in  the  order  in  which  they  were  obtained, 
but  in  the  ascending  order  of  the  diameters,  and  I  have  distributed 
them  into  groups  of  analogous  diameters.  During  the  continuance 
of  the  operations  the  temperature  varied  from  18°*5  to  20°  C. 

*  *  *  *  As  the  first  diameter  is  to  those  of  the  last  group 
very  nearly  as  1  to  6,  these  results  suffice,  I  think,  to  establish  dis- 
tinctly the  constancy  of  the  product  p  d,  and  consequently  the  law 
according  to  which  the  pressure  is  in  the  inverse  ratio  of  the  dia- 
meter. 

********* 

"As  to  the  general  mean  22'75  of  the  results  of  the  table,  its  deci- 
mal part  is  necessarily  a  little  too  high,  on  account  of  the  excessive 


Diameters,  or 
values  of  d. 

Pressures,  or 
values  of  p. 

Products,   or 
values  of  p  d. 

m  m 

mm 

7-55 

3-00 

22-65 

10-37 

2-17 

22-50 

10-55 

2-13 

22-47 

23-35 

0-98 

22-88 

26-44 

0-83 

21-94 

27-58 

0-83 

22-89 

46-60 

0-48 

22-37 

47-47 

0-48 

22-78 

47-85 

0-43 

20-57 

48-10 

0-55 

26-45 

108  Lecture- Notes  on  Physics. 

value  26-45  of  the  last  product.  As  this  product,  and  that  which 
precedes  it,  are  those  which  alone  deviate  materially  from  22  in 
this  integral  part,  it  will  be  admitted,  I  think,  that  a  nearer  approach 
to  the  true  value  will  be  made  by  neglecting  these  two  products  and 
taking  the  mean  of  the  others,  a  mean  which  is  22*56,  or  more  sim- 
ply 22-6 ;  we  shall  adopt,  then,  this  last  number  for  the  value  of 
the  product  p  d  in  regard  to  the  gly eerie  liquid. 

It  remained  to  be  verified  whether  this  value  satisfied  our  for- 
mula, according  to  which  we  have  p  d  =  2  h  S-,  the  quantities  £•  and 
h  being  respectively,  as  we  have  seen,  the  density  of  the  liquid  and 
the  height  which  this  liquid  would  attain  in  a  capillary  tube  1 
millimetre  in  diameter.  With  this  view,  therefore,  it  was  necessary 
to  seek  the  values  of  these  two  quantities  in  reference  to  the  glyce- 
ric  liquid.  The  density  was  determined  by  means  of  the  aerometer 
of  Fahrenheit,  at  the  temperature  of  17°  C.,  a  temperature  little 
inferior  to  that  of  the  preceding  experiments,  and  the  result  was 
>  =  1-1065.  To  determine  the  capillary  height  the  process  of  Gay 
Lussac  was  employed,  that  is,  the  measurement  by  the  cathetometer, 
all  known  precautions  being  taken  to  secure  an  exact  result.  The 
experiment  was  made  at  the  temperature  of  19°  C.  **•*•* 
*  *  •*  rp]^  reading  of  the  cathetometer  gave,  for  the  distance 
from  the  lowest  point  of  the  concave  meniscus  to  the  exterior  level, 
27mm-35. 

This  measurement  having  been  taken,  the  tube  was  removed, 
cut  at  the  point  reached  by  the  capillary  column,  and  its  interior 
diameter  at  that  point  measured  by  means  of  a  microscope,  furnished 
with  a  micrometer,  giving  directly  hundredths  of  a  millimetre.  It 
was  found  that  the  interior  section  of  the  tube  was  slightly  elliptical, 
the  greater  diameter  being  Omm-374  and  the  smaller  Omm'357;  the 
mean  was  adopted,  namely,  Omm'3655,  to  represent  the  interior  dia- 
meter of  the  tube  assumed  to  be  cylindrical.  To  have  the  true 
height  of  the  capillary  column,  it  is  necessary,  we  know,  to  add  to 
the  height  of  the  lowest  point  of  the  meniscus  the  sixth  part  of  the 
diameter  of  the  tube,  or,  in  the  present  case,  Omm>06  ;  the  true  height 
of  our  column  is  consequently  27miu>41.  Now,  to  obtain  the  height 
h  to  which  the  same  liquid  would  rise  in  a  tube  having  an  interior 
diameter  of  exactly  a  millimetre,  it  is  sufficient,  in  virtue  of  the 
known  law,  to  multiply  the  above  height  by  the  diameter  of  the 
tube,  and  thus  we  find  definitively  &  =  10min-018. 

"I  should  here  say  for  what  reason  I  have  chosen  for  the  experi- 


Lecture-Notes  on  Physics.  109 

ment  a  tube  whose  interior  diameter  is  considerably  less  than  a 
millimetre.  The  reasoning  by  which  I  arrived  at  the  formula  sup- 
poses that  the  surface  which  terminates  the  capillary  column  is 
hemispherical ;  now  that  is  not  strictly  true,  but  in  a  tube  so  nar- 
row as  that  which  I  have  employed,  the  difference  is  wholly  inap- 
plicable, so  that  in  afterwards  calculating,  by  the  law  of  the  inverse 
ratio  of  the  elevation  to  the  diameter,  the  height  for  a  tube  one  mil- 
limetre in  diameter,  we  would  have  this  height  such  as  it  would  be 
if  the  upper  surface  were  exactly  hemispherical. 

uThe  values  of  £  and  h  being  thus  determined,  we  deduce  there- 
from 2/0  =  22-17,  a  number  which  differs  but  little  from  22'56 
obtained  above  as  the  value  of  the  product  p  d.  The  formula 
pd=2h§  may  therefore  be  regarded  as  verified  by  experiment, 
and  the  verification  will  appear  still  more  complete  if  we  consider 
that  the  two  results  are  respectively  deduced  from  elements  altogether 
different.  I  hope  hereafter  to  obtain  new  verifications  with  other 
liquids. 

Investigation  of  a  very  small  limit  below  which  is  found,  in  the 
gly  eerie  liquid,  the  value  of  the  radius  of  sensible  activity  of  the  mole 
cular  attraction. 

2  h& 
"  The  exactness  of  the  formula  p  =  — -v-  supposes,  as  we  are  about 

to  show,  that  the  film  which  constitutes  the  bubble  has,  at  all  points, 
no  thickness  less  than  double  the  radius  of  sensible  activity  of  the 
molecular  attraction. 

"We  have  seen  that  the  pressure  exerted  by  a  bubble  on  the  air 
which  it  encloses  is  the  sum  of  the  actions  separately  due  to  the 
curvatures  of  its  two  faces.  On  the  other  hand,  we  know  that, 
in  the  case  of  a  full  liquid  mass,  the  capillary  pressure  exerted  by 
the  liquid  on  itself  emanates  from  all  points  of  a  superficial  stratum 
having  as  its  thickness  the  radius  of  activity  in  question.  Now,  if 
the  thickness  of  the  film  which  constitutes  a  bubble  is  everywhere 
superior  or  equal  to  double  that  radius,  each  of  the  two  faces  of  the 
film  will  have  its  superficial  stratum  unimpaired,  and  the  pressure 
exerted  on  the  enclosed  air  will  have  the  value  indicated  by  our 
formula.  But  if,  at  all  its  points,  the  film  has  a  thickness  inferior 
to  or  double  this  radius,  the  two  superficial  strata  have  not  their 
complete  thickness,  and  the  number  of  molecules  comprised  in  each 
of  them  being  thus  lessened,  these  two  strata  must  necessarily  exert 


110  Lecture- Notes  on  Physics. 

actions  less  strong,  and  consequently  the  sum  of  these,  that  is  to 
say,  the  pressure  on  the  interior  air,  must  be  smaller  than  the  for- 
mula indicates  it  to  be.  Hence  it  follows  that  if,  in  the  experiments 
described  above,  the  thickness  of  the  films  which  formed  the  bub- 
bles had,  through  the  whole  extent  of  these  last,  descended  below 
the  limit  in  question,  the  results  would  have  been  too  small,  but  in 
this  case  we  should  have  remarked  progressive  and  continued 
diminutions  in  the  pressures,  which,  however,  never  happened, 
although  the  color  of  the  bubbles  evinced  great  tenuity.  But  all 
physicists  admit  that  the  radius  of  sensible  activity  of  the  molecular 
attraction  is  excessively  minute. 

"But  what  precedes  permits  of  our  going  further,  and  deducing 
from  experiment  a  datum  on  the  value  of  the  radius  of  sensible 
activity,  at  least  in  the  glyceric  liquid. 

#-3f*##### 

"After  the  film  has  acquired  a  uniform  thinness,  if  the  pressure 
exerted  on  the  air  within  the  bubble  underwent  a  diminution,  this 
would  be  evinced  by  the  manometer,  and  it  would  be  seen  to  progress 
in  a  continuous  manner  in  proportion  to  the  ulterior  attenuation  of 
the  film.  In  this  case  the  thickness  which  the  film  had  when  the  di- 
minution of  pressure  commenced  would  be  determined  by  the  tinge 
which  the  central  space  presented  at  that  moment,  and  the  half  of 
that  thickness  would  be  the  value  of  the  radius^ of  sensible  activity 
of  the  molecular  attraction.  If,  on  the  contrary,  the  pressure  re- 
mains constant  until  the  disappearance  of  the  bubble,  we  may  infer 
from  the  tint  of  the  central  space  the  final  thickness  of  the  film,  and 
the  half  of  this  thickness  will  constitute  at  least  a  limit,  very  little 
below  which  is  to  be  found  the  radius  in  question. 

•fc#-fc#-5f-fc4f# 

*  *  *  I  deposited  in  the  bottom  of  the  dry  jar,  morsels  of 
caustic  potash,  and  contrived  by  the  application  of  a  little  lard 
around  the  orifice  of  the  jar  and  of  the  aperture  through  which 
passed  the  copper  tube,  that  after  the  introduction  of  the  bubble, 

the  pasteboard  disk  should  close  the  opening  hermetically.     *     * 
##-*####* 

"Now,  under  these  conditions,  the  diminution  of  thickness  of  the 
film  was  continuous,  the  bubble  lasted  for  nearly  three  days,  and 
when  it  burst,  it  had  arrived  at  the  transition  from  the  yellow  to 
the  white  of  the  first  order ;  it  then  presented  a  central  space  of  a 
pale  yellow  tint,  surrounded  by  a  white  ring. 


Lecture- Notes  on  Physics.  Ill 

•***•*•*•*<*#• 

*  *  *  if  the  pressure  varied,  it  was  in  an  irregular  manner, 
in  both  directions,  and  terminating  not  in  a  diminution,  but  an 
augmentation,  at  least  relative;  we  may,  therefore,  admit,  I  think, 
that  the  final  thickness  of  the  film  was  still  superior  to  double  the 
radius  of  sensible  activity  of  the  molecular  attraction. 

"Let  us  now  see  what  we  may  deduce  from  this  last  experiment. 
According  to  the  table  given  by  Newton,  the  thickness  of  a  film 
of  pure  water  which  reflects  the  yellow  of  the  first  order  is,  in  mil- 
lionths  of  an  English  inch,  5J,  or  5'333,  and  for  the  white  of  the 
same  order  3 {•,  or  3*875.  We  may  therefore  take  the  mean,  namely 
4-064,  as  the  closely  approximative  value  of  the  thickness  corres- 
ponding, at  least  in  the  case  of  pure  water,  to  the  transition  between 
those  colors,  and  the  English  inch  being  equal  to  25-4  millimetres, 
this  thickness  is  equivalent  to  55^4  of  a  millimetre.  Now  we  know 
that,  for  two  different  substances,  the  thickness  of  the  films  which 
reflect  the  same  tint  is  in  the  inverse  ratio  of  the  indices  of  refrac- 
tion of  those  substances.  In  order,  therefore,  to  obtain  the  real 
thickness  of  our  film  of  glyceric  liquid,  it  suffices  to  multiply  the 
denominator  of  the  preceding  fraction  by  the  ratio  of  the  index  of 
the  glyceric  liquid  to  that  of  water.  I  have  measured  the  former 
flpproximatively  by  means  of  a  hollow  prism,  and  have  found  it 
equal  to  1-377.  That  of  water  being  1-336,  there  results,  for  the 
thickness  of  the  glyceric  film  g^T  of  a  millimetre.  The  half  of  this 
quantity,  or  77-552  of  a  millimetre,  constitutes,  therefore,  the  limit 
furnished  by  the  experiment  in  question.  Hence  we  arrive  at  the 
very  probable  conclusion,  that  in  the  glyceric  liquid  the  radius  of 
sensible  activity  of  the  molecular  attraction  is  less  than  I^^-Q-Q  of  a 
n.illimetre. 

"I  had  proposed  to  continue  this  investigation  with  a  view  to 
reach,  if  possible,  the  black  tint,  and  to  elucidate  the  variations  of 
the  manometer ;  but  the  cold  season  has  intervened,  diminishing 
the  persistence  of  the  bubbles,  and  I  have  been  forced  to  postpone 
attempts  to  a  more  favorable  period." 

In  the  report  of  the  transactions  of  the  Society  of  Physics  and 
Natural  History,  of  Geneva,  1862,  we  find  the  following :  "  Prof. 
Wartmann,  Jr.,  repeated  before  the  Society  the  recent  experiments 
of  M.  Plateau  on  bubbles  of  soap,  of  varied  forms  as  well  as  much 
persistency,  obtained  by  mixing  with  soap-suds  a  small  quantity  of 
glycerine,  and  causing  the  bubbles  to  attach  themselves  to  iron 


112  Lecture-Notes  on  Physics. 

wires  arranged  in  different  manners.  At  a  subsequent  session  M. 
Wartmann  exhibited  an  apparatus  of  the  same  kind,  still  more 
varied,  so  as  to  produce  more  perfectly  than  by  former  processes 
the  phenomena  of  coloration  in  extremely  thin  surfaces  of  the  liquid. 
The  dark  part  presents  not  more  than  y^Vsu  of  a  millimetre,  whence 
we  may  conclude,  says  M.  Wartmann,  that  the  radius  of  the  sensi- 
ble activity  of  molecular  attraction  is  below  w?\w  °f  a  milli- 
metre." 


ADDITIONS  AND  CORRECTIONS. 


Page  3,  line  11  from  top,  for  such  are  the  planets,  &c.,  read  such 
are  the  stars,  the  sun,  the  planets  and  asteroids  of,  &c. 

Page  3,  after  line  21,  read:  Nebulas  are  faintly  luminous  aggre 
gations  of  matter  so  far  removed  from  our  solar  system 
as  to  have  no  sensible  parallax.  They  are  formed 
either  of  clusters  of  stars,  and  are  then  resolvable,  or 
qonsist  of  self-luminous  gaseous  matter,  and  are  then 
unresolvable  into  star  points ;  while  some  seem  formed 
of  stars  surrounded  by  atmospheres  of  intensely  heated 
vapors,  as  is  the  case  with  our  sun.  These  facts  are  the 
results  of  the  spectroscopic  analysis  of  these  bodies.  If 
the  spectrum  of  a  celestial  body  is  formed  only  of  iso- 
lated bright  lines,  then  that  body  is  composed  only  of 
luminous  gas,  as  is  the  case  with  the  great  nebula  in 
Orion.  When  the  spectrum  is  continuous,  the  light 
comes  from  an  incandescent  solid  or  liquid  mass;  while 
if  the  spectrum  is  continuous  and  crossed  by  fine  dark 
lines,  we  know  that  the  light  emanates  from  an  incan- 
descent solid  or  liquid  body  surrounded  by  vapors 
which  have,  by  their  absorption,  produced  these  dark 
lines. 

Of  60  nebulas  examined  by  Mr.  Huggins,  the  spectroscope 
showed  that  20  were  gaseous,  while  40  consisted  of  ag- 
gregations of  stars. 
(See  Results  of  Spectrum  Analysis  Applied  to  the  Heavenly 

Bodies.     By  Wm.  Huggins.     London:  1868.) 
The  Zodiacal   light   is   a   faint   nebulous    light,    resem- 
bling that  of  the  tail  of  a  comet.     It   is   visible   in 
the    W.  about   1st   March,    after    sunset,    and   in  the 


E.  about  the  middle  of  October,  before  sunrise.  It  is 
supposed  to  be  caused  by  a  nebulous  envelope  which 
surrounds  the  sun  and  extends  beyond  the  orbit  of  the 
earth,  so  that,  if  the  sun  were  seen  from  one  of  the  stars 
exterior  to  our  system,  it  would  appear  as  a  bright 
point  enveloped  in  a  haze,  just  as  some  of  the  fixed 
stars  appear  to  us. 

Page  4,  line  3  from  top,  for  solids,  liquids  and  gases,  read  solid, 
liquid  and  gaseous. 

Page  4,  line  8  from  bottom,  for  stone,  associated  facts.  (1)  Spaces 
read  stone.  Associated  facts,  (1)  spaces. 

Page  17,  line  8  from  top,  for  TJ^  read  O1IJU. 

Page  18,  line  7  from  top,  after  audience,  read,  by  the  aid  of  a 
lantern  provided  with  a  small  slit  and  a  lens. 

Page  19,  after  line  3  from  top,  read:  See  papers  on  Robert's 
Test-plates,  by  Mr.  Stodder,  in  the  American  Naturalist 
for  April,  1868,  and  by  Messrs.  Sullivant  and  Wood- 
ward in  Amer.  Jour.  Science,  Nov.,  1868. 

Page  20,  after  line  17  from  top,  read:  An  arc  of  17°  10',  of  which 
the  cord  is  J  of  radius,  may  be  employed  as  a  test  of 
the  accuracy  of  the  work. 

Page  24,  line  2  from  top,  for  millimetre  read  milligramme. 

Page  24,  line  16  from  bottom,  for  mean  revolution,  read  mean 
apparent  revolution. 

Page  25,  line  1  from  top,  for  dead-beat,  read  dead  beat  and  gravity. 

Page  25,  after  line  4  from  bottom,  read  Dr.  Locke,  of  Cincinnati, 
in  1848,  first  recorded  transits  by  means  of  an  electric 
clock  and  the  ordinary  Morse  telegraph  register;  .and 
Messrs.  Saxton  and  Bond  subsequently  replaced  the 
telegraph  fillet  by  a  uniformly  revolving  cylinder* 
This  mode  of  registering  transits  is  known  as  "the 
American  method." 

Page  28,  add  to  the  table  Litre  =   0-035317  cubic  feet, 

"    =61-02705      "•    inches. 

Page  31  and  32.  In  this  discussion  we  have  assumed  that  the 
arithmetical  mean  is  the  mean  of  an  infinite  number 
of  observations,  and,  therefore,  that  its  error  is  an  infi- 
nitesimal ;  but,  as  the  number  of  observations  is  always 
limited,  better  approximations  to  the  values  of  the 


probable  errors  of  a  series  of  directly  observed  quanti- 
ties are  given  by  the  formulas : — 


n—l 
Probable  error  of  a  single  observation  is — 

0-6745  e 
Probable  error  of  the  mean  result  is — 

E  =  0-674:5 - =0-6745  \J  *2  +  *  /2  +t//2  + 

Vn  n(n— 1) 

Page  35,  line  11  from  bottom,  for  effected  read  affected. 
Page  66,  line  8  from  top,  for  height  read  heights. 
Page  72,  line  11  from  top,  for  E  and  F  is,  read  E  and  F  are. 
Page  81,  line  22  from  top,  dele  and. 

Page  87,  line  11  from  bottom,  for  of  the  ring  read  at  the  ring. 
Page  90,  line  19  from  top,  for  Tyndall  and  others  read  Tyndall, 
Henry,  Norton  and  others. 


Entered,  according  to  the  Act  of  Congress,  in  the  year  1869,  by 

ALFRED  M.  MAYER,  Ph.D., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  in  and  for  the 
Eastern  District  of  Pennsylvania. 


°  ° °* 


419 


